This passage contains teaching in verses 12-14 about the Pharisees as "blind guides of the blind," but let's save our discussion of that section for next time. Right now we'll look at the discussion between Jesus and Peter on what truly defiles a man.
The tradition of hand-washing before eating isn't written in Old Testament law. Most likely the Pharisees took the commandment for priests to wash before ministering at the altar of the temple (Exodus 30:17-21) and applied it universally to all Jews coming before the table to eat a meal. It's a way of ramping up the holiness of the nation, and maybe then God will be pleased to restore Israel to her former glory. Of course, not only were the Pharisees teaching a man-made tradition as if it were the commandment of God, but they were violating a true commandment of God, the fifth commandment, in the process.
But Jesus' criticism of the hand-washing tradition reaches far beyond the fact that it is a mere tradition. He attacks the entire rationale upon which it is based. He says it's not what you put in your mouth that defiles you, it's what comes out of your mouth straight from your corrupt heart that defiles you. What is shocking about this pronouncement is that Jesus is challenging the entire idea of clean and unclean distinctions as taught by the Mosaic Law. The parallel passage in Mark 7:1-23 even adds as an aside, "Thus [Jesus] declared all foods clean," though Matthew seems to be content with just planting that idea in our heads and letting us draw our own conclusions. The law delineated categories of clean and unclean to train the Israelites to think in these stark, black-and-white terms, but the law was only a tutor, an intensive training exercise to help God's people see that the true clean and unclean distinction is between a holy God and sinful men.
Jesus rebukes the disciples for their lack of understanding, because anyone with spiritual insight would realize that the real uncleanness the law is referring to lies within our hearts. Food has nothing to do with it. Our words betray how much evil is hidden inside each one of us, an unending stream of lust, deceit and murder. Jesus had to come to us as the clean one and take our uncleanness upon himself. He became unclean to God, rejected upon the cross, so that we could be declared clean in the Father's sight.