Properly applying the law: Murder -
Jesus introduces the proper applications of the law that he talked about in the previous verses (v.17-20). Rather than overthrowing the OT law, he is properly interpreting it. We see this most clearly from two facts:
1) He does not begin, "it is written," but rather "you have heard." Elsewhere he points to Scripture with, "it is written." (Matthew 4, Matthew 21:13)
2) He doesn't say anything the Old Testament doesn't already state. Leviticus 19:17 - "You shall not hate your brother in your heart."
Given that, we are affirmed in believing that instead of overthrowing the Law of the Bible, he is properly interpreting it.
A plausible conjecture is that the teachers of the law were saying that as long as you didn't actually commit murder, you weren't guilty of breaking the law. You could insult them and hate them, but they were not guilty of murder.
In a complete rejection of that interpretation, Jesus says, "but I say to you... To even hate your brother meant you were guilty of murder. To insult them meant you were liable to the council, to call them a fool meant you were in danger of the fires of hell." To Jesus, and to God's people, there should not be a line drawn between the physical act of murder and the spiritual inclination to it.
God doesn't just want our external actions. Just because I haven't committed the physical act of murder doesn't mean that I was innocent. Like the scribes and pharisees, my danger is to interpret the law by the letter and let myself off the hook. That's not what God commands - he doesn't want just the action, but the heart also.