After the reading of the foregoing epistle, the most reverend bishops cried out: This is the faith of the fathers, this is the faith of the Apostles. So we all believe, thus the orthodox believe. Anathema to him who does not thus believe. Peter has spoken thus through Leo. So taught the Apostles. Piously and truly did Leo teach, so taught Cyril. Everlasting be the memory of Cyril. Leo and Cyril taught the same thing, anathema to him who does not so believe. This is the true faith. Those of us who are orthodox thus believe. This is the faith of the fathers. Why were not these things read at Ephesus [i.e. at the heretical synod held there]? These are the things Dioscorus hid away.(source)
Those who are trying to argue for an ancient view of papal primacy love to interpret this as a recognition by the council of Leo's unique Petrine authority. In other words, they would interpret this as meaning "Leo is today's Peter."
But the tone of the comments is somewhat puzzling, if they think Leo is the pope that must be obeyed. It looks as though they are blessing and giving approval to Leo's comments. They are authorizing him, rather than vice versa.
Moreover, there is a much more logical and straightforward explanation for their words. Leo's letter had argued in this way:
But when our Lord and Saviour himself was by his questions instructing the faith of the disciples, he said, “Whom do men say that I the Son of Man am?” And when they had mentioned various opinions held by others, he said, “But whom say ye that I am?” that is, “I who am Son of Man, and whom you see in the form of a servant, and in reality of flesh, whom say ye that I am?” Whereupon the blessed Peter, as inspired by God, and about to benefit all nations by his confession, said, “Thou art the Christ, the Son of the living God.” Not undeservedly, therefore, was he pronounced blessed by the Lord, and derived from the original Rock that solidity which belonged both to his virtue and to his name, who through revelation from the Father confessed the selfsame to be both the Son of God and the Christ; because one of these truths, accepted without the other, would not profit unto salvation, and it was equally dangerous to believe the Lord Jesus Christ to be merely God and not man, or merely man and not God.(source)
Let him listen also to the blessed Apostle Peter when he declares, that “sanctification by the Spirit” takes place through the “sprinkling of the blood of Christ,” and let him not give a mere cursory reading to the words of the same Apostle, “Knowing that ye were not redeemed with corruptible things, as silver and gold, from your vain way of life received by tradition from your fathers, but with the precious blood of Jesus Christ as of a Lamb without blemish and without spot.”
As an aside, note that Leo identifies "the original Rock" as Christ here. The main point of providing this selection, though, is to highlight the more obvious source of the words of these men. Leo argues from the words of Peter, both in Matthew and in Peter's first epistle. In this way, Peter spoke through Leo, in that Leo repeated Peter's words. It's the most obvious and straightforward explanation of their comment.
I think it is worth pointing out what Leo thought about the role of Scripture in the issue of resolving the heresy that they were dealing with:
But into this folly do they fall who, when hindered by some obscurity from apprehending the truth, have recourse, not to the words of the Prophets, not to the letters of the Apostles, nor to the authority of the Gospels, but to themselves; and become teachers of error, just because they have not been disciples of the truth. For what learning has he received from the sacred pages of the New and the Old Testament, who does not so much as understand the very beginning of the Creed? And that which, all the world over, is uttered by the voices of all applicants for regeneration, is still not grasped by the mind of this aged man. If, then, he knew not what he ought to think about the Incarnation of the Word of God, and was not willing, for the sake of obtaining the light of intelligence, to make laborious search through the whole extent of the Holy Scriptures, he should at least have received with heedful attention that general Confession common to all, whereby the whole body of the faithful profess that they “believe in God the Father Almighty, and in Jesus Christ his only Son our Lord, who was born of the Holy Ghost and the Virgin Mary.”And, of course, that's what the rest of his letter does - it goes through the Scriptures and demonstrates the proof of the matter, including the testimony of Peter in Matthew and Peter's Epistle, as well as the testimony of John in 1 John:
Nor has he been overawed by the declaration of the blessed Apostle and Evangelist John, saying, “Every spirit that confesseth that Jesus Christ has come in the flesh is of God; and every spirit which dissolveth Jesus is not of God, and this is Antichrist.” Now what is to dissolve Jesus, but to separate the human nature from him, and to make void by shameless inventions that mystery by which alone we have been saved?So, if you happen to find Roman apologists quoting this to support the authority of the papacy (as one sees at Catholic Answers) or in Jimmy Akin's new book (sorry, no preview available), even if you happen to find it used in the more nuanced approach of Klaus Schatz or the rather extreme position of Susan Wessel, just bear in mind the context. Understand that Leo's letter was based on Scripture and was accepted because of the authority of Scripture.
Let him also not resist the testimony of Blessed John the Apostle, “And the blood of Jesus the Son of God cleanseth us from all sin.” And again, “This is the victory which overcometh the world, even our faith;” and, “who is he that overcometh the world, but he that believeth that Jesus is the Son of God? This is he that came by water and blood, even Jesus Christ; not in water only, but in water and blood; and it is the Spirit that beareth witness, because the Spirit is truth. For there are three that bear witness—the Spirit, the water, and the blood; and the three are one.” That is, the Spirit of sanctification, and the blood of redemption, and the water of baptism; which three things are one, and remain undivided, and not one of them is disjoined from connection with the others; because the Catholic Church lives and advances by this faith, that Christ Jesus we should believe neither manhood to exist without true Godhead, nor Godhead without true manhood.
That is the faith of fathers - that is what Leo taught - that is what we believe.