I made a post on the Catholic Answers forums recently on this subject, and since it was "blog-length," :) I thought it fitting to include here with some minor touch-ups for improvement.
Let's look at John 6:63
John 6:63 The Spirit gives life; the flesh counts for nothing. The words I have spoken to you are spirit and they are life.
Those who interpret this as Jesus canceling out all His prior literal language in the chapter in favor of a symbolic understanding of the Eucharist run into a problem. First, if their interpretation were correct, to be consistent they would have to say Christ's sacrifice of blood on the cross counts for nothing.
But second, that is not what Jesus meant by "the words I have spoken are spirit." Consider the flow of the text in John 6 (which by the way comes right after 2 miracles: multiplication of loaves & fishes and walking on water).
Jesus' audience in John 6 is a signpost for the Real Presence. It's actually important to know that Jesus often spoke symbolically of Himself, like when He said: "I am the door" (John 10:7), or "you are the salt" (Mat. 5:13), or "take my yoke" (Mat. 11:29). Since the disciples knew Jesus sometimes spoke symbolically, it makes no sense that they would suddenly take "symbolic" words literally if He was indeed speaking symbolically yet again. They would have naturally said, "Oh! He's speaking symbolically of course! As He often does!" (EDIT 12/15/10: See later in John an example of Jesus clarifying Himself when His disciples actually did misunderstand him John 11:11-14.)
But the audience understood Jesus literally. "How can He give us His flesh to eat?" they said (John 6:52) and "This is a hard saying; who can listen to it?" (John 6:60). Jesus does not reply, "Because I'm speaking symbolically" or the like. Instead, He responds, "Do you take offense at this?" (John 6:61). And the gospel writer adds: "Jesus knew from the first who those were that did not believe" (John 6:64). These untrusting disciples departed from Jesus over this issue. And Jesus let them walk away thinking He had spoken literally.
If a person insists Jesus was speaking symbolically, then that person as a byproduct makes Jesus a deceiver for seeing they did not understand Him correctly and still letting them depart from Him. Jesus actually confirmed their literal understanding when He did not correct them but rather asked: "Does this offend you?"
Of course, those departing disciples correctly understood Jesus literally. And Jesus knew it. Peter, who was there, did not understand how Jesus could be speaking literally either---but he knew by faith to trust Jesus. "To whom shall we go?" (John 6:68) Peter said when asked by Jesus if he also wished to depart. Peter's faith preceded his understanding. Such as it is with the mystery of the Eucharist. But the point is, not a single member of Jesus' audience, whether those who departed nor those who remained, understood Him to be speaking symbolically. And Jesus confirmed their understanding in His responses.
So what does "the words I have spoken are spirit" mean from John 6:63? It means His words were spoken such that they are understood not by the mind of "flesh." In other words---His words are not understood by human reason, a natural understanding. Rather, only a spiritual person can understand how Jesus can give us His literal flesh and blood to eat.
I think St. John Chrysostom, ca 390 A.D., explains this verse well:
His meaning is, 'Ye must hear spiritually what relateth to Me, for he who heareth carnally is not profited. (Homilies on John)
Christ's words are spiritual words, not carnal words. Spiritual does not equal "symbolic."
Jesus made a similar statement earlier in the gospel of John:
John 3:12 If I have told you earthly things and you do not believe, how can you believe if I tell you heavenly things?Here we see the same contrast between the carnal and the spiritual with regard to understanding what Jesus says.
Consider two final passages from Paul that tie into this:
1 Corinthians 2:14 The unspiritual man does not receive the gifts of the Spirit of God, for they are folly to him, and he is not able to understand them because they are spiritually discerned.1 Corinthians 11:28-29 Let a man examine himself, and so eat of the bread and drink of the cup. For any one who eats and drinks without discerning the body eats and drinks judgment upon himself.
You see how Paul teaches the exact same thing about discerning Christ's body in the Eucharist by using a spiritual understanding?