To answer the question, it is important to understand that Catholics (as well as the Eastern Orthodox and other eastern churches) make a sharp distinction between adoration (or worship), which is directed only to God, and veneration, which refers to the honor given to the saints.
When Catholics (and Orthodox) pray to Mary and the other saints, it is never seen as an act of adoration, but only of veneration. It should be noted that Catholics (and Orthodox) are also encouraged to pray to God directly; in fact, for them prayer in its various facets (praise, thanksgiving, petition, supplication, and adoration) is chiefly directed to Him, not to saints (and whenever it involves adoration, it is only directed to God).
Therefore, the prayers that Catholics (and Orthodox) make to saints are chiefly for their intercession. Intercessory prayer is something that St. Paul clearly encouraged among the living:
First of all, then, I urge that supplications, prayers, intercessions, and thanksgivings be made for all people, for kings and all who are in high positions, that we may lead a peaceful and quiet life, godly and dignified in every way. This is good, and it is pleasing in the sight of God our Savior, who desires all people to be saved and to come to the knowledge of the truth (1 Tim. 2:1-4, emphasis added).
St. Paul also asked people to pray for him, as is pointed out by Geremia:
I appeal to you, brothers, by our Lord Jesus Christ and by the love of the Spirit, to strive together with me in your prayers to God on my behalf (Rom 15:30).
St. Paul, however, also teaches that saints are in a more perfect union with their Creator than we are:
For now we see in a mirror dimly, but then face to face. Now I know in part; then I shall know fully, even as I have been fully known (1 Cor. 13:12).
If those who have still not entered into the fullness of union with God (“for all have sinned and fall short of the glory of God,” Rom. 3:23) can intercede for one another, then all the more those who are already in possession of that glory can intercede for those on earth.
In fact, intercessory prayer is quite commonly recommended in the Scriptures. Moses had the privilege of speaking to God face-to-face:
Thus the Lord used to speak to Moses face to face, as a man speaks to his friend (Ex. 33:11).
If an Israelite needed to ask a favor from God, he would always go through Moses. Certainly, the ancient Israelites had only a partial understanding of God and His goodness, and so they feared to approach God directly, but this imperfect understanding does not take away the value of intercessory prayer. The intercessory role assumed by Moses was continued by the Levitical priests throughout the history of the People of Israel. (The whole Book of Leviticus describes this role in detail.)
The Scriptures, then, clearly encourage intercessory prayer, and it does not forbid us to seek this intercession from the saints in heaven. Moreover, it seems reasonable (according to Catholics and Orthodox) to seek the intercession of those that are closest to God.
It is true that the Scripture does not make an explicit endorsement of the practice, but demanding an explicit endorsement of every practice is a rather restrictive criterion. (For example, the Bible does not explicitly endorse revival meetings, altar calls, or other typically Protestant practices.) Nevertheless, there are indications in the Scriptures that the early Church firmly believed in the intercession of the saints. Probably the best example can be found in Rev. 5:8:
And when he had taken the scroll, the four living creatures and the twenty-four elders fell down before the Lamb, each holding a harp, and golden bowls full of incense, which are the prayers of the saints.
In this context the “saints” (hagioi) are actually the members of the Church living on earth. In this passage, St. John (the author of Revelation) takes for granted those who are before the throne of the Lamb (i.e., God the Son) have the task of presenting the prayers (incense) of those on earth. This is intercession on the part of heavenly creatures on behalf of those still living on earth. (The 24 elders probably represent the 12 patriarchs of Israel and the 12 Apostles; hence John is most likely referring here to human, not angelic, intercessors.)
Let us now examine some possible objections to the Catholic (and Orthodox) practice of praying to the saints:
Aren’t the saints dead? Didn’t God forbid contact with the dead?
The saints are “dead” in the sense that they are no longer physically present on earth, but they are, in fact, very much alive in Heaven, in perfect union with their Creator. Jesus himself makes this very point when answering the Sadduccees:
And as for the resurrection of the dead, have you not read what was said to you by God: ‘I am the God of Abraham, and the God of Isaac, and the God of Jacob’? He is not God of the dead, but of the living (Mt. 22:32).
It is true that God prohibits necromancy, or conjuring the dead:
There shall not be found among you anyone who burns his son or his daughter as an offering, anyone who practices divination or tells fortunes or interprets omens, or a sorcerer or a charmer or a medium or a necromancer or one who inquires of the dead, for whoever does these things is an abomination to the Lord. And because of these abominations the Lord your God is driving them out before you. You shall be blameless before the Lord your God, for these nations, which you are about to dispossess, listen to fortune-tellers and to diviners. But as for you, the Lord your God has not allowed you to do this (Deut. 18:10-14).
Necromancy means obliging a dead man’s soul to appear on earth by magical incantations. Praying to saints for their intercession has nothing to do with this practice. Necromancy, unlike prayer to the saints, does not respect the freedom of the one who is conjured up (even assuming that it is successful: there is always the grave risk of conjuring up an evil spirit instead of the dead person in question).
In summary: the saints are not dead, nor does praying to them constitute a forbidden, occult practice.
How can the saints hear us?
Although Rev. 5:8 indicates that those before the throne of the Lamb have knowledge of our prayers, a possible objection is that the saints, mere creatures, are not omniscient, and so they cannot hear all of our prayers.
Catholics (and Orthodox) would reply that the saints, being creatures, are certainly not omniscient; that is, they do not have all knowledge, as their Creator does. They do, however, enjoy the direct vision of God, and this vision entails knowledge of all those things that are relevant to them.
Moreover, time in Heaven does not pass in the same way as time on earth:
But do not overlook this one fact, beloved, that with the Lord one day is as a thousand years, and a thousand years as one day (2 Peter 3:8).
For a thousand years in your sight are but as yesterday when it is past, or as a watch in the night (Ps. 90:4).
Since it is through their vision of God that the saints acquire their knowledge, they have no problem hearing our prayers, even if those prayers are numerous.
Isn’t Jesus the One Mediator?
A possible objection stems from the following passage in 1 Timothy:
For there is one God, and there is one mediator between God and men, the man Christ Jesus (1 Tim. 2:5).
Catholics (and Orthodox) do not dispute the uniqueness of Christ’s mediation, which is based on the very Incarnation. Jesus’ unique status as the One Mediator does not, they would say, prevent God from establishing lesser mediations subordinate to the first. Jesus, as it were, bridges the gap, opened by the sin of Adam, between God and man; that does not prevent him from establishing other “bridges” between himself and the men still on earth. Jesus, in fact, clearly did this by establishing the Church, and the Apostles to lead it.
In any case, St. Paul cannot be forbidding intercessory prayer (which is mediation of the secondary kind), because he has just commanded it in the immediately preceding passage. (See above.)
Can’t we pray to Jesus directly?
Catholics (and Orthodox) would emphatically reply in the affirmative. Not only can we pray to Jesus directly, we should do so, and doing so should be the main focus of our prayer. They would say that intercessory prayer (including asking saints for their prayers) is simply one of the ways that God wishes to involve all members of the Church in the work of redemption: after all, if we ask the saints to pray for us, it is so that they they (as well as we) pray to Jesus.