The following is from Jonathan Edwards’s notes on the Bible, and deals specifically with Matthew 27:51.
“And, behold, the veil of the temple was rent in twain from the top to the bottom.” This was the veil that hindered our access to the throne of grace, or the mercy-seat in the holy of holies.
That hiding of the mercy-seat, and hindering of our access to the mercy-seat, figured a twofold hinderance of access to God.
1. The hinderance by which all men are kept off while they remain under the first covenant of works; they are hindered by their guilt, God’s law and justice.
2. That hinderance of free access that was under the first testament while the church was in a legal state, and in its minority, and under carnal, ordinances, so that access was rare and difficult, only allowed to the high priest, and that but once a year: so that the veil signifies two things, viz. 1. The sin of man, both guilt and corruption of heart; which both in diverse respects are a veil to hide the mercy-seat, and hinder our access. Both these were typified by the flesh of Christ. The sin of God’s people, or elect church, was typified by Christ’s flesh; for sin is called flesh in Scripture, and the elect church is Christ mystical; so that Christ, in taking flesh upon him, took their sin upon him: he became sin for us, and when his flesh was crucified, when his human nature died, then this veil was removed, for that abolished the sin of the elect church. So likewise Christ in the flesh, in his infirm, weak state, signified the church, or Christ mystical, in its Old-Testament minority, when it was in its weak, infirm, and carnal state, under carnal ordinances, under the elements of the world; and those carnal ordinances, and carnal dispensations, that Christ mystical was under, was as it were the flesh of Christ. When Christ died, then there was an end to those types and shadows, because they were then all fulfilled.
Christ’s human nature was a temple; it was the antitype of the temple; and his flesh, or the infirmity of his human nature, was the veil that hid the glory of God, or the divinity that dwells in him, and was in his person. So that the veil of the temple, in the third place, typified the literal flesh of Christ, that had veiled his glory; which it ceased to do when his state of humiliation was at an end. Christ himself, our great High Priest, entered into the holy of holies through the veil of his own flesh. That day that Christ died, was the great day of atonement, typified by the day of atonement of old, when the high priest entered into the holy of holies. Christ, as God man, could enter into heaven no other way than by rending this veil. Christ offered his sacrifice in the outward court, in this world, and then in the conclusion of it rent the veil, that his blood might be sprinkled within the veil. [source]