In this answer to another question, a protestant view is given on what it means that the temple veil was torn. I have started to do some quick research into how the Catholic Church has understood it, but have mostly found the words of modern apologists and don't have time to do extensive research. This given interpretation does not ring true in my heart and obviously does not reflect Catholic teaching so that is the reason for my question.
In order to understand what the Catholic Church teaches about the tearing of the veil, we must have a proper understanding of the Catholic priesthood's relativity to the Temple of the Old Covenant.
There are three types of priesthood in the Catholic Church: (1) Priesthood of Christ as the High Priest and true Priest, the only mediator between God and men; (2) ministerial (or hierarchical) priesthood comprising bishops and priests and (3) priesthood of all faithful though baptism.
1544 Everything that the priesthood of the Old Covenant prefigured finds its fulfillment in Christ Jesus, the "one mediator between God and men." The Christian tradition considers Melchizedek, "priest of God Most High," as a prefiguration of the priesthood of Christ, the unique "high priest after the order of Melchizedek"; "holy, blameless, unstained," "by a single offering he has perfected for all time those who are sanctified," that is, by the unique sacrifice of the cross.
1545 The redemptive sacrifice of Christ is unique, accomplished once for all; yet it is made present in the Eucharistic sacrifice of the Church. The same is true of the one priesthood of Christ; it is made present through the ministerial priesthood without diminishing the uniqueness of Christ's priesthood: "Only Christ is the true priest, the others being only his ministers."
Two participations in the one priesthood of Christ
1546 Christ, high priest and unique mediator, has made of the Church "a kingdom, priests for his God and Father." The whole community of believers is, as such, priestly. The faithful exercise their baptismal priesthood through their participation, each according to his own vocation, in Christ's mission as priest, prophet, and king. Through the sacraments of Baptism and Confirmation the faithful are "consecrated to be . . . a holy priesthood."
1547 The ministerial or hierarchical priesthood of bishops and priests, and the common priesthood of all the faithful participate, "each in its own proper way, in the one priesthood of Christ." While being "ordered one to another," they differ essentially. In what sense? While the common priesthood of the faithful is exercised by the unfolding of baptismal grace --a life of faith, hope, and charity, a life according to the Spirit--, the ministerial priesthood is at the service of the common priesthood. It is directed at the unfolding of the baptismal grace of all Christians. The ministerial priesthood is a means by which Christ unceasingly builds up and leads his Church. For this reason it is transmitted by its own sacrament, the sacrament of Holy Orders. (Catechism of the Catholic Church # 1544 to 1547)
While accepting the first and the third ones, Protestant and other post-Reformation churches have rejected the ministerial priesthood for two reasons:
New Testament does not apply the title priest (Greek iereus) to the apostles, bishops (Greekepiskopos) and elders (Greek presbuteros).
The Eucharist or Lord’s Supper or Holy Communion, is only a remembrance of Christ, following his request (Luke 22:19) – it does not have sacrificial nature as believed by Catholics.
1330 The memorial of the Lord's Passion and Resurrection.
The Holy Sacrifice, because it makes present the one sacrifice of Christ the Savior and includes the Church's offering. The terms holy sacrifice of the Mass, "sacrifice of praise," spiritual sacrifice, pure and holy sacrifice are also used, since it completes and surpasses all the sacrifices of the Old Covenant.
The Holy and Divine Liturgy, because the Church's whole liturgy finds its center and most intense expression in the celebration of this sacrament; in the same sense we also call its celebration the Sacred Mysteries. We speak of the Most Blessed Sacrament because it is the Sacrament of sacraments. The Eucharistic species reserved in the tabernacle are designated by this same name. (Catechism of the Catholic Church #1330)
Catholics believe the priesthood of the Old Covenant is the one that prefigures that of the New. First the Priesthood of Christ is prefigured by that of Melchizedek.
And Melchizedek king of Salem brought out bread and wine; now he was a priest of God Most High. (Genesis 14:18)
In the same way, Christ did not take on himself the glory of becoming a high priest. But God said to him, “You are my Son; today I have become your Father.” And he says in another place, “You are a priest forever, in the order of Melchizedek.” During the days of Jesus’ life on earth, he offered up prayers and petitions with fervent cries and tears to the one who could save him from death, and he was heard because of his reverent submission. Son though he was, he learned obedience from what he suffered and, once made perfect, he became the source of eternal salvation for all who obey him and was designated by God to be high priest in the order of Melchizedek. (Hebrews 5:5-10)
The priesthood of all the elect in the New Covenant is prefigured by all of God's chosen Israelites of the Old Covenant.
Now if you obey me fully and keep my covenant, then out of all nations you will be my treasured possession. Although the whole earth is mine, you will be for me a kingdom of priests and a holy nation.’ These are the words you are to speak to the Israelites.” (Exodus 19:5-6)
As you come to him, the living Stone—rejected by humans but chosen by God and precious to him— you also, like living stones, are being built into a spiritual house to be a holy priesthood, offering spiritual sacrifices acceptable to God through Jesus Christ. But you are a chosen people, a royal priesthood, a holy nation, God’s special possession, that you may declare the praises of him who called you out of darkness into his wonderful light. (1 Peter 2:4-5, 9)
To him who loves us and has freed us from our sins by his blood, 6and has made us to be a kingdom and priests to serve his God and Father—to him be glory and power for ever and ever! Amen. (Revelation 1:5-6)
In addition, God chose Aaron and his sons to be His priests, and appointed them and the tribe of Levi (to which Aaron belonged) for liturgical service in the Sanctuary.
“Have Aaron your brother brought to you from among the Israelites, along with his sons Nadab and Abihu, Eleazar and Ithamar, so they may serve me as priests. Make sacred garments for your brother Aaron to give him dignity and honor. Tell all the skilled workers to whom I have given wisdom in such matters that they are to make garments for Aaron, for his consecration, so he may serve me as priest. (Exodus 28:1-3)
The LORD said to Aaron, “You, your sons and your family are to bear the responsibility for offenses connected with the sanctuary, and you and your sons alone are to bear the responsibility for offenses connected with the priesthood. Bring your fellow Levites from your ancestral tribe to join you and assist you when you and your sons minister before the tent of the covenant law. They are to be responsible to you and are to perform all the duties of the tent, but they must not go near the furnishings of the sanctuary or the altar. Otherwise both they and you will die. They are to join you and be responsible for the care of the tent of meeting—all the work at the tent—and no one else may come near where you are. (Number 18:1-4)
These were the divisions of the descendants of Aaron: The sons of Aaron were Nadab, Abihu, Eleazar and Ithamar. But Nadab and Abihu died before their father did, and they had no sons; so Eleazar and Ithamar served as the priests. With the help of Zadok a descendant of Eleazar and Ahimelek a descendant of Ithamar, David separated them into divisions for their appointed order of ministering. 4A larger number of leaders were found among Eleazar’s descendants than among Ithamar’s, and they were divided accordingly: sixteen heads of families from Eleazar’s descendants and eight heads of families from Ithamar’s descendants. They divided them impartially by casting lots, for there were officials of the sanctuary and officials of God among the descendants of both Eleazar and Ithamar. (1 Chronicles 24:1-5)
Thus priests and Levites, whenever these two appear in the Bible refer to these two groups specially chosen by God.
They brought up the ark of the LORD and the tent of meeting and all the holy utensils, which were in the tent, and the priests and the Levites brought them up. (1 Kings 8:4)
Then Hezekiah questioned the priests and the Levites concerning the heaps. (2 Chronicles 31:9)
For the priests and the Levites had purified themselves together; all of them were pure. Then they slaughtered the Passover lamb for all the exiles, both for their brothers the priests and for themselves. (Ezra 6:20)
Likewise we cast lots for the supply of wood among the priests, the Levites and the people so that they might bring it to the house of our God, according to our fathers' households, at fixed times annually, to burn on the altar of the LORD our God, as it is written in the law; (Nehemiah 10:34)
This is the testimony of John, when the Jews sent to him priests and Levites from Jerusalem to ask him, "Who are you?" (John 1:19)
The Sanctuary is the place where God was present with Israelites.
"Let them construct a sanctuary for Me, that I may dwell among them. (Exodus 25:8)
We read how God instructed Moses to build a tent to function as the Sanctuary in the book of Exodus chapters 25-27.
Later God chose Solomon to build the first Temple in Jerusalem to house the Sanctuary.
In the four hundred and eightieth year after the Israelites came out of Egypt, in the fourth year of Solomon’s reign over Israel, in the month of Ziv, the second month, he began to build the temple of the LORD. (1 Kings 6:1)
"Consider now, for the LORD has chosen you to build a house for the sanctuary; be courageous and act." (1 Chronicles 28:10)
The Sanctuary consisted of 2 parts separated by a veil the (outer) Holy Place and the (inner) Holy of Holies.
“Make a curtain of blue, purple and scarlet yarn and finely twisted linen, with cherubim woven into it by a skilled worker. Hang it with gold hooks on four posts of acacia wood overlaid with gold and standing on four silver bases. Hang the curtain from the clasps and place the ark of the covenant law behind the curtain. The curtain will separate the Holy Place from the Most Holy Place. (Exodus 26:31-33)
A tabernacle was set up. In its first room were the lampstand and the table with its consecrated bread; this was called the Holy Place. Behind the second curtain was a room called the Most Holy Place, which had the golden altar of incense and the gold-covered ark of the covenant. This ark contained the gold jar of manna, Aaron’s staff that had budded, and the stone tablets of the covenant. Above the ark were the cherubim of the Glory, overshadowing the atonement cover. (Hebrews 9:2-5)
Only priests were allowed to offer sacrifice in the (outer) Holy Place. One of the priests will be appointed as High Priest and only he can enter the (inner) Holy of Holies and only once every year to offer blood atonement
"Aaron shall make atonement on its horns once a year; he shall make atonement on it with the blood of the sin offering of atonement once a year throughout your generations. It is most holy to the LORD." (Exodus 30:10)
When everything had been arranged like this, the priests entered regularly into the outer room to carry on their ministry. But only the high priest entered the inner room, and that only once a year, and never without blood, which he offered for himself and for the sins the people had committed in ignorance. (Hebrews 9:6-7)
Catholics believe that the priesthood of Melchizedek prefigures that of Christ, while the priesthood of Aaron and his sons prefigures that of bishops, institution of seventy elders prefigures that of priests (the English word priest is derived from Greek presbuteros), the Levites prefigures the deacons while priesthood of all Israelites prefigures the common priesthood of all faithful.
1540 Instituted to proclaim the Word of God and to restore communion with God by sacrifices and prayer, this priesthood nevertheless remains powerless to bring about salvation, needing to repeat its sacrifices ceaselessly and being unable to achieve a definitive sanctification, which only the sacrifice of Christ would accomplish.
1544 Everything that the priesthood of the Old Covenant prefigured finds its fulfillment in Christ Jesus, the "one mediator between God and men." The Christian tradition considers Melchizedek, "priest of God Most High," as a prefiguration of the priesthood of Christ, the unique "high priest after the order of Melchizedek"; "holy, blameless, unstained," "by a single offering he has perfected for all time those who are sanctified," that is, by the unique sacrifice of the cross. (Catechism of the Catholic Church #1540, 1544)
Most, if not all, Protestants and “Bible only” Christians will argue that priesthood of Aaron or Levitical priesthood is abolished in New Testament. Yet Scripture prophesies that the Levitical priesthood shall never lack a man in God’s presence to offer burnt offerings, to burn cereal offerings, and to make sacrifice forever.
For this is what the LORD says: ‘David will never fail to have a man to sit on the throne of Israel, nor will the Levitical priests ever fail to have a man to stand before me continually to offer burnt offerings, to burn grain offerings and to present sacrifices.’ ” (Jeremiah 33:18)
The Veil Is Torn
Christ established the New Covenant (Matthew 26:28, Luke 22:20, 1 Corinthians 11:25) on the night he was betrayed. The old covenant refers to the one God made with Moses, and it is to be replaced with the new and better covenant of Christ (Hebrews 8:6), prophesied in Jeremiah 31:33-34 (Hebrews 8:8-12, 10:16-17). The Sanctuary of the New Covenant is heavenly and not man-made - it is heaven itself (Hebrews 8:5, 9:11, 24).
In the Old Covenant priests offered sacrifice in the outer Sanctuary. The Veil separated it from the inner Sanctuary into which only the High Priest could enter (prefigured Christ). The torn veil as testified in Matthew 27:51 shows that in the New Covenant there is no longer separation between the sacrifice offered by priests and Christ, the High Priest.
All Catholics receive their common priesthood through Sacrament of Baptism and Confirmation.
1546 Christ, high priest and unique mediator, has made of the Church "a kingdom, priests for his God and Father." The whole community of believers is, as such, priestly. The faithful exercise their baptismal priesthood through their participation, each according to his own vocation, in Christ's mission as priest, prophet, and king. Through the sacraments of Baptism and Confirmation the faithful are "consecrated to be . . . a holy priesthood." (Catechism of the Catholic Church # 1546)
Christ, being the the only mediator between God and men (1 Timothy 2:5), is the only true Priest. Both ministerial and common priesthood participate in this one priesthood of Christ.
1547 The ministerial or hierarchical priesthood of bishops and priests, and the common priesthood of all the faithful participate, "each in its own proper way, in the one priesthood of Christ." While being "ordered one to another," they differ essentially. In what sense? While the common priesthood of the faithful is exercised by the unfolding of baptismal grace --a life of faith, hope, and charity, a life according to the Spirit--, the ministerial priesthood is at the service of the common priesthood. It is directed at the unfolding of the baptismal grace of all Christians. The ministerial priesthood is a means by which Christ unceasingly builds up and leads his Church. For this reason it is transmitted by its own sacrament, the sacrament of Holy Orders. (Catechism of the Catholic Church # 1546-1547)
The New American Bible, Revised Edition presents an orthodox summary of Matthew’s account of the tearing of the veil:
Luke puts this event immediately before the death of Jesus. There were two veils in the Mosaic tabernacle on the model of which the temple was constructed, the outer one before the entrance of the Holy Place and the inner one before the Holy of Holies (see Ex 26:31–36). Only the high priest could pass through the latter and that only on the Day of Atonement (see Lv 16:1–18). Probably the torn veil of the gospels is the inner one. The meaning of the scene may be that now, because of Jesus’ death, all people have access to the presence of God, or that the temple, its holiest part standing exposed, is now profaned and will soon be destroyed. The earth quaked…appeared to many: peculiar to Matthew. The earthquake, the splitting of the rocks, and especially the resurrection of the dead saints indicate the coming of the final age. In the Old Testament the coming of God is frequently portrayed with the imagery of an earthquake (see Ps 68:9; 77:19), and Jesus speaks of the earthquakes that will accompany the “labor pains” that signify the beginning of the dissolution of the old world (Mt 24:7–8). For the expectation of the resurrection of the dead at the coming of the new and final age, see Dn 12:1–3. Matthew knows that the end of the old age has not yet come (Mt 28:20), but the new age has broken in with the death (and resurrection; cf. the earthquake in Mt 28:2) of Jesus; see note on Mt 16:28. After his resurrection: this qualification seems to be due to Matthew’s wish to assert the primacy of Jesus’ resurrection even though he has placed the resurrection of the dead saints immediately after Jesus’ death. (USCCB)