Those who believe in penal substitution mean something along on the lines of this: that Christ was made to have human nature. This was done in order to take the penalty of the law upon himself for other humans (not for angels) as a substitutionary sacrifice. By suffering on their behalf the wrath of God and just punishments of his law are extinguished, provisioning forgiveness for any who accept that work on their behalf by faith.
For example Charles Hodge focuses on the penal aspect when discussing atonement:
The Scriptures, however, assume that if a man sins he must die. On this assumption all their representations and arguments are founded. Hence the plan of salvation which the Bible reveals supposes that the justice of God which renders the punishment of sin necessary has been satisfied. Men can be pardoned and restored to the favour of God, because Christ was set forth as an expiation for their sins, through faith in his blood; because He was made a curse for us; because He died, the just for the unjust; because He bore our sins in his own body on the tree; and because the penalty due to us was laid on Him. It is clear, therefore, that the Scriptures recognize the truth that God is just, in the sense that He is determined by his moral excellence to punish all sin, and therefore that the satisfaction of Christ which secures the pardon of sinners is rendered to the justice of God. Its primary and principal design is neither to make a moral impression upon the offenders themselves, nor to operate didactically on other intelligent creatures, but to satisfy the demands of justice; so that God can be just in justifying the ungodly. (SYSTEMATIC THEOLOGY, CHARLES HODGE, v2.492)
Or expressing the wider idea more vividly we can sample one of many texts from Luther's commentary on Galatians:
And so this text is clear, that all men, even the apostles or prophets or patriarchs, would have remained under the curse if Christ had not put Himself in opposition to sin, death, the curse of the Law, and the wrath and judgment of God, and if He had not overcome them in His own body; for those savage monsters could not be overcome by any human power. Now Christ is not the Law, He is not a work of the Law, He is not an “elicited act”; but He is a divine and human Person who took sin, the condemnation of the Law, and death upon Himself, not for Himself but for us. Therefore the whole emphasis is on the phrase ὑπὲρ ἡμῶν.
Therefore we should not imagine Christ as an innocent and private person who is holy and righteous only for Himself; this is what the sophists and nearly all the fathers, Jerome and others, have done. It is, of course, true that Christ is the purest of persons; but this is not the place to stop. For you do not yet have Christ, even though you know that He is God and man. You truly have Him only when you believe that this altogether pure and innocent Person has been granted to you by the Father as your High Priest and Redeemer, yes, as your Slave. Putting off His innocence and holiness and putting on your sinful person, He bore your sin, death, and curse; He became a sacrifice and a curse for you, in order thus to set you free from the curse of the Law. (Luther's Works V26 p287)
Now regarding the sacrifices of the Old Covenant, people of this faith (i.e. reformed faith) have always considered those sacrifices as symbolic representations of the future sacrifice of Christ. Therefore when offering those sacrifices it was not merely a factual prophecy (as opposed to a prophecy through words) but a religious rite that could be used to express saving faith in Christ alone. However, many did not have this faith under that covenant and were therefore not actually forgiven (almost everyone, for example, in the first generation after Egypt were slain for unbelief in the desert). But even for the wicked who still followed the ceremony prior to Christ, the sin offerings were figuratively a 'substitute for the penalties of sin'.
That is the background of the whole idea. To go directly to your point I think the answer is found in those core sacrifices related to the forgiveness of sins. The point you would find interested in and which answers your question is that the person offering the sacrifice would put his hands on the head of the beast transferring his guilt and sin onto the animal being put to death for those sins.
There are many sources to go to to verify this undisputed fact. I like Alfred Edersheim, an old Jewish historian and academic. In his book 'THE TEMPLE, ITS MINISTRY AND SERVICES AS THEY WERE AT THE TIME OF JESUS CHRIST' he speaks about the 'laying on of hands':
This meant transmission and delegation, and implied representation; so that it really pointed to the substitution of the sacrifice for the sacrificer. Hence it was always accompanied by confession of sin and prayer. It was thus done. The sacrifice was so turned that the person confessing looked towards the west, while he laid his hands between the horns of the sacrifice, and if the sacrifice was brought by more than one, each had to lay on his hands. It is not quite a settled point whether one or both hands were laid on; but all are agreed that it was to be done ‘with one’s whole force’—as it were, to lay one’s whole weight upon the substitute. If a person under vow had died, his heir-at-law took his place. The only public sacrifices in which hands were laid on were those for sins of public ignorance, when the ‘elders’ acted as representing the people—to which some Rabbinical authorities add public sin-offerings in general,—and the scapegoat on the Day of Atonement, on which the high-priest laid his hands. In all private sacrifices, except firstlings, tithes, and the Paschal lamb, hands were laid on, and, while doing so, the following prayer was repeated: ‘I entreat, O Jehovah: I have sinned, I have done perversely, I have rebelled, I have committed (naming the sin, trespass, or, in case of a burnt-offering, the breach of positive or negative command); but I return in repentance, and let this be for my atonement (covering).’ (P116)
Of course the practice of the temple has no valid meaning unless derived from scriptural practice, so it is worth noting an example of where the same practice originated:
“When Aaron has finished making atonement for the Most Holy Place, the tent of meeting and the altar, he shall bring forward the live goat. 21 He is to lay both hands on the head of the live goat and confess over it all the wickedness and rebellion of the Israelites—all their sins—and put them on the goat’s head. He shall send the goat away into the wilderness in the care of someone appointed for the task. 22 The goat will carry on itself all their sins to a remote place; and the man shall release it in the wilderness. (Lev 16:20)
The formal word generally used for the result made by this substitution is atonement.
11 For the life of a creature is in the blood, and I have given it to you to make atonement for yourselves on the altar; it is the blood that makes atonement for one’s life. (NIV Leviticus 17:11)
One must realize this need to be reconciled to God for our sin through substitutionary sacrifice is natural as it was derived from paradise in the shedding of animal skins to cover their nakedness. It started before Moses, but it's typical nature of Christ, when only under Moses was formally laid down in law.
The basic fact is the blood of another life is shed for your blood, symbolic of one life being substituted for yours. The type finding its antitype in Christ, where his soul is sacrificed for yours. If this is not substitutionary I don't know what possibly could be?!
The result of laying ones hands on the beast (transference of sin) , the subsequent death of the beast (substituted shedding of life blood for yours) is atonement or 'covering'.
1 Blessed is the one whose transgressions are forgiven, whose sins are covered. (NIV Psalm 32:1)
Covering (atonement) is the result of the substitution. Or as the historian of Jewish history says:
This idea of substitution, as introduced, adopted, and sanctioned by God Himself, is expressed by the sacrificial term rendered in our version ‘atonement,’ but which really means covering, the substitute in the acceptance of God taking the place of, and so covering, as it were, the person of the offerer. (THE TEMPLE ITS MINISTRY AND SERVICES AS THEY WERE AT THE TIME OF JESUS CHRIST, ALFRED EDERSHEIM, p107)
So you see the sacrifices were as a 'substitutionary-symbol'. They symbolically covered the sins of a guilty party by shedding the blood of another. Of course only as a symbol of the sacrifice of a future Messiah, for "It is impossible for the blood of bulls and goats to take away sins. (Heb 10:1).
The above part has no conclusive argumentative power unless there is an important assumption maintained: death is penal in and of itself. death is punishment from the law for breaking its threatening demand. (of course by faith in Christ the otherwise penal nature of death upon all men is removed Heb 2:15 but that is the exception to the eternal rule.)
One would be hard pressed to find a theologian holding the penal view, that actually explicitly states this assumption when explaining the meaning of sacrifices as it is deemed self evident and unnecessary to say! Death would not be existent if it were not for sin! Therefore, death is a punishment of breaking God’s law to Adam as clearly as the Sun is a source of heat and light. Therefore, when we speak of a ‘sin’ offering (sin being the cause of all punishment and death) we intend to mean an offering with respect to punishment due for sin. In fact, to the penal view removing the assumption that 'death is a penalty' would make the bible one gigantic and absurd riddle.
However, as it has come to my attention that this assumption, thought to be obvious by reformed theologians, is not assumed by all mankind. Let me just merely show then that this IS the assumption of all those who are reformed and when applied to Part one of the answer, explains the reformed view authentically.
John Owen is probably the most famous theologian in history that most fully argues the penal view of death in all his many writings:
The word death is penal
. 2. Death in the first constitution of it was penal. 3. It is still penal, eternally penal, to all unbelievers. 4. The death of all is equally determined and certain in God’s constitution. 5. The ground of the expiation of sin by the offering of Christ is this, that therein he bare the guilt and punishment due unto it. Owen, J. (n.d.). The works of John Owen. (William H. Goold, Ed.) (Vol. 19, pp. 504–505).
Physical death, sin, loss of righteousness, spiritual death, eternal death; are all Penal
The death wherewith he was threatened stood in opposition to all these, it being most ridiculous to suppose that any thing penal in the Scripture comes under the name of “death” that was not here threatened to Adam;—death of the body, in a deprivation of his immortality spoken of; of the soul spiritually, in sin, by the loss of his righteousness and integrity; of both, in their obnoxiousness to death eternal; actually to be undergone, without deliverance by Christ, in opposition to the right to a better, a blessed condition, which he had. That all these are penal, and called in the Scriptures by the name of “death,” is evident to all that take care to know what is contained in them. Owen, J. (n.d.). The works of John Owen. (William H. Goold, Ed.) (Vol. 12, p. 151).
Penal death are best represented as painful, bloody, associated with sin, as were the bloody sacrifices of the Aaronic priesthood
Of death natural, which in its whole nature is penal (as hath been elsewhere evinced), there are four aggravations, whereunto all others may be referred: as,—(1.) That it be violent or bloody; (2.) That it be ignominious or shameful; (3.) That it be lingering and painful; (4.) That it be legal and accursed. And all these to the height met in the death of Christ. The works of John Owen. (William H. Goold, Ed.) (Vol. 12, p. 485).
Even some Jewish Rabbis have recognized the penal nature of all death
The Jews themselves grant that all death is penal: אין מות בלא חטא ואין ייסורין בלא עוין;—“There is no death without sin, no punishment or correction without iniquity.” It is the saying of R. Ame in the Talmud, Tractat. Sabbat., cited in Sepher Ikharim, lib. iv. cap. xiii. And this principle Maimonides carries so high as to deny all יסורין של אחבה, “correction of love,” affirming none to be of that mind but some Gaeonims, deceived by the sect of Muatzali, More Nebuch. pag. 3, cap. xvii. And they who die penally under the curse abide in no other estate than that mentioned. They acknowledge, also, the remainder of the curse on the earth itself on the same account: העולם כלו לא נברא אלא בשביל האדם ואחר שאדם חטא האדמה חסרה שלמותה;—“The whole world,” says one of their masters, “was not created but for man; and therefore after man sinned, it came short of its first perfection.” An Exposition of the Epistle to the Hebrews. (W. H. Goold, Ed.) (Vol. 18, p. 148).
An Expert of Jewish Customs and historical beliefs recognises the sacrifice as penal death
The fundamental idea of sacrifice in the Old Testament is that of substitution, which again seems to imply everything else—atonement and redemption, vicarious punishment and forgiveness. The firstfruits go for the whole products; the firstlings for the flock; the redemption-money for that which cannot be offered; and the life of the sacrifice, which is in its blood, for the life of the sacrificer. Edersheim, A. (1959). The Temple, its ministry and services as they were at the time of Jesus Christ. (p. 107).
Combining Part1 one and Part2 here is the summary argument:
- Death is punishment for breaking God’s law (i.e. death is penal).
- Although some innocent things die for Adam’s sin such as vegetables and trees, this is still as punishment on humanity and is penal.
- When an animal is specifically set aside as a sin offering, the transference of sin mandates transference of punishment, otherwise no transfer has occurred. The soul that sin’s shall die if that sin be upon him, so if that sin be upon another so shall it die if it is to be logical. One can’t separate sin from death, and death is penal.
- Sin offerings under the Old Testament were both symbols and rites. Symbols; of transference of bloody painful guilty death/punishments, whereby Christ was prefigured. Also a practical rites; to ceremonially cleanse a worshiper so that they could continue to live under the covenant for lesser sins, through ignorance etc.
- Sin offerings could not even ceremonially cleanse those sins that the Law demanded capital punishment for, such as adultery, murder etc. They could not actually spiritually cleanse anything but were symbols (Acts 13:39).
- As symbols however, believers could express faith in Christ and actually be cleansed from sin through faith alone. In that way the symbol acts as the word to believe in.
- Jews someone times did not have faith in the symbols and were often not cleansed spiritually by them in the above manner, so although they practiced the ceremony and were clean for worship they were still under the penal death of the law.
- The reason for 'many-many' symbols in the Old Covenant is that they were all weak in representing the full picture of the anti-type (Christ). Even all the sacrifices combined could not even 'ceremonially' cleanse a single sin requiring capital punishment. The weakness in the type does not mean the anti-type could not bear that punishment for that sin. The sin of the whole world could be forgiven when he cried out 'Why have you forsaken me?' in his agonising penal death.
Death is obviously penal. Death is a punishment for sin from the Law that originally threatened to punish Adam if he did sin. Sin and penal death are inseparable joined. If it can be proven that sin is transferred to the sacrifice then penal death has also been transferred. If death has been transferred; punishment has been transferred, as death is penal. Sin offerings are therefore obviously symbolic substitutions of penal-death for sin under the law, to be employed ceremonially by believers of the old covenant. (when the Law did not already prescribe capital punishment for the sin in question). These symbolic and ceremonial substitutions of penal death for sin, had a primary purpose above that mere ceremonial cleansing allowing worshipers to continue temple practice: They were a ‘type of Christ’ prophesying the future role of the Messiah who would absorb the penal death of Adam’s curse under God's eternal threatening Law. This great sacrifice would take away even those sins that could not be cleaned under the ceremonial system such as murder.(Acts 13:39)
It is worth noting that under the question of ancient sacrifices, one can’t logically avoid thinking of Christ in his bloody, guilty and painful punishment, to deliver those under the penal Law. One can't be accused on adding something additional to the subject. Everything in the Old Testament is somewhat cloudy and vague in comparison to the bright light of the gospel. Therefore the clarity of the old shadows are not fully understood but through the lens of the gospel. So the interpretation of 'if the sacrifices under the Old Testament are penal' or not, is actually just casting one’s view of 'if Christ's death was penal' or not. The type and ant-type must match and be used to interpret each other. If death is penal from Adam, and Christ carries that penal death, then the symbolic sacrifices that pointed to him must be penal. However, even before Christ came a good theologian would have to have understood the penalty in the symbolism associated with penal death. A good theologian must recognise death is penal even before the New Testament was written.
Comment 1. One person said he did not think that the death that Adam was punished with for sinning against his law was 'punishment' but merely a 'consequence'. The soul that sins shall die, is just a bad consequence.
Answer 1: A sinner who is under the compulsion of a sinful nature commits adultery is strangled to death under the punishment of God's Old Testament Law. Yet Adam, whose sin is the greatest ever, as he ushered all humanity into death by his sin without even being under the compulsion of a sinful nature to temp him inwardly, only 'consequentially' suffers death? That is backward.