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The Catechism rarely explains the Bible verse by verse, but you can read the Catholic teaching in its entirety here:

This is the collection of patristic literature on that particular verse.

CHRYS. He tries to remove their difficulties in another way, as follows, It is the spirit that quickens, the flesh profits nothing: that is to say, You ought to understand My words in a spiritual sense: he who understands them carnally is profited nothing. To interpret carnally is to take a proposition in its bare literal meaning, and allow no other. But we should not judge of mysteries in this way; but examine them with the inward eye; i.e. understand them spiritually. It was carnal to doubt how our Lord could give His flesh to eat. What then? Is it not real flesh? Yes, verily. In saying then that the flesh profits nothing, He does not speak of His own flesh, but that of the carnal hearer of His word

AUG. Or thus, the flesh profits nothing. They had under stood by His flesh, as it were, of a carcass, that was to be cut up, and sold in the shambles, not of a body animated by the spirit. Join the spirit to the flesh, and it profits much: for if the flesh profited not, the Word would not have become flesh, and dwelt among us. The Spirit has done much for our salvation, by means of the flesh.

AUG. For the flesh does not cleanse of itself, but by the Word who assumed it: which Word, being the principle of life in all things, having taken up soul and body, cleanses the souls and bodies of those that believe. It is the spirit, it, then, that quickens: the flesh profits nothing; i.e. the flesh as they understood it. I do not, He seems to say, give My body to be eaten in this sense. He ought not to think of the flesh carnally: The words that I speak to you, they are spirit, and they are life.

CHRYS. i.e. are spiritual, have nothing carnal in them, produce no effects of the natural sort; not being under the dominion of that law of necessity, and order of nature established on earth.

AUG. If then you understand them spiritually, they are life and spirit to you: if carnally, even then they are life and spirit, but not to you. Our Lord declares that in eating His body, and drinking His blood, we dwell in Him, and He in us. But what has the power to affect this, except love? The love of God is shed abroad in our hearts by the Holy Spirit, which is given to us.

Catena Aurea John 6

Clement of Alexandria wrote,

Further release from evils is the beginning of salvation. We then alone, who first have touched the confines of life, are already perfect; and we already live who are separated from death. Salvation, accordingly, is the following of Christ: For that which is in Him is life. John 1:4 "Verily, verily, I say unto you, He that hears My words, and believes in Him that sent Me, has eternal life, and comes not into condemnation, but has passed from death to life." John 5:24 Thus believing alone, and regeneration, is perfection in life; for God is never weak. For as His will is work, and this is named the world; so also His counsel is the salvation of men, and this has been called the church. He knows, therefore, whom He has called, and whom He has saved; and at one and the same time He called and saved them...

As nurses nourish new-born children on milk, so do I also by the Word, the milk of Christ, instilling into you spiritual nutriment..."Wherefore also I have given you milk to drink," he says; meaning, I have instilled into you the knowledge which, from instruction, nourishes up to life eternal. But the expression, "I have given you to drink" (ἐπότισα), is the symbol of perfect appropriation. For those who are full-grown are said to drink, babes to suck. "For my blood," says the Lord, "is true drink." John 6:55 In saying, therefore, "I have given you milk to drink," has he not indicated the knowledge of the truth, the perfect gladness in the Word, who is the milk? And to this meaning we may secondly accommodate the expression, "I have given you milk to drink, and not given you food, for you are not yet able," regarding the meat not as something different from the milk, but the same in substance. For the very same Word is fluid and mild as milk, or solid and compact as meat. And entertaining this view, we may regard the proclamation of the Gospel, which is universally diffused, as milk; and as meat, faith, which from instruction is compacted into a foundation, which, being more substantial than hearing, is likened to meat, and assimilates to the soul itself nourishment of this kind.

Elsewhere the Lord, in the Gospel according to John, brought this out by symbols, when He said: Eat my flesh, and drink my blood; John 6:34 describing distinctly by metaphor the drinkable properties of faith and the promise, by means of which the Church, like a human being consisting of many members, is refreshed and grows, is welded together and compacted of both—of faith, which is the body, and of hope, which is the soul; as also the Lord of flesh and blood. For in reality the blood of faith is hope, in which faith is held as by a vital principle. And when hope expires, it is as if blood flowed forth; and the vitality of faith is destroyed. ” (Clement of Alexandria, The Paedagogus, Book I; http://www.newadvent.org/fathers/02091.htm)

Tertullian:

Now, because they thought His discourse was harsh and intolerable, supposing that He had really and literally enjoined on them to eat his flesh, He, with the view of ordering the state of salvation as a spiritual thing, set out with the principle, “It is the spirit that quickens;” and then added, “The flesh profits nothing,”— meaning, of course, to the giving of life. He also goes on to explain what He would have us to understand by spirit: “The words that I speak unto you, they are spirit, and they are life.” In a like sense He had previously said: “He that hears my words, and believes in Him that sent me, has everlasting life, and shall not come into condemnation, but shall pass from death unto life.” John 5:24 Constituting, therefore, His word as the life-giving principle, because that word is spirit and life, He likewise called His flesh by the same appellation; because, too, the Word had become flesh, John 1:14 we ought therefore to desire Him in order that we may have life, and to devour Him with the ear, and to ruminate on Him with the understanding, and to digest Him by faith. (On the Resurrection of the Flesh, Chapter 37; http://www.newadvent.org/fathers/0316.htm)

And Augustine on "Rule for Interpreting Commands and Prohibitions" states,

24. If the sentence is one of command, either forbidding a crime or vice, or enjoining an act of prudence or benevolence, it is not figurative. If, however, it seems to enjoin a crime or vice, or to forbid an act of prudence or benevolence, it is figurative. "Unless you eat the flesh of the Son of man," says Christ, "and drink His blood, you have no life in you." John 6:53 This seems to enjoin a crime or a vice; it is therefore a figure, enjoining that we should have a share [communicandem] in the sufferings of our Lord, and that we should retain a sweet and profitable memory [in memoria] of the fact that His flesh was wounded and crucified for us. Augustine On Christian Doctrine (Book III, cp. 16) — http://www.newadvent.org/fathers/12023.htm