"Do not think that I have come to abolish the law or the prophets; I have come not to abolish but to fulfill. For truly I tell you, until heaven and earth pass away, not one letter, not one stroke of a letter, will pass from the law until all is accomplished. Therefore, whoever breaks one of the least of these commandments, and teaches others to do the same, will be called least in the kingdom of heaven; but whoever does them and teaches them will be called great in the kingdom of heaven. For I tell you, unless your righteousness exceeds that of the scribes and Pharisees, you will never enter the kingdom of heaven." (Matthew 5:17-20) (NRSV)
With these words, Jesus moves into the body of the Sermon on the Mount. He appears to stake out a position, possibly a claim. Most of the remainder of the chapter provides illustrates how he applies the claim to specific situations (see 5:21-47).
Christian commentators generally note that Jesus continues to deal with the kingdom of heaven. In this section, he speaks of its connection to the law and the prophets. Whatever may be said (and people at different places on the Christian theological specrum indeed have said many and varied things), Jesus did not intend to abolish the law and the prophets. Instead, he claims to "fulfill them."
That's interesting, if for no other reason, because few Christians since the late first or early second century CE have thought it neccesary to observe dietary, ceremonial, or civil laws found in the Hebrew Bible. For the most part, they argue that all such laws were "fulfilled" in some way via Jesus' sacrificial death. Most then go on to insist that the "moral" law (personal behavior, etc.) remains in force, now and forever. Personally, I'm not particularly enamored of this approach. It feels more like a philosopher's approach to truth (slicing something into various parts and applying labels) than the wholistic approach to life, which I believe typical of the Hebrew Bible and of Jesus.
I try to take the phrase "the law and the prophets" seriously. Both the law and the teachings of the prophets had to do with making a life pleasing to God and hence good for the individual, the community and even the larger world. Perhaps the best way to read "but to fulfill" might be "but to enflesh" or "but to make clear in all their radicality." The remainder of the section and chapter makes clear that Jesus thought the law and prophets called for something more than most people (in his view) thought.
At the very least, Jesus raised the bar. While they have often gotten bad press among Christians, the Pharisees seem to have been generally well regarded by many first century Jews. In short, many if not most everyday Jews would have seen a Pharisee's brand of righteousness as something to be emulated or at least admired as a high standard. Verse 20, I think, would have come as a bit of a shock to Jesus' audience.
What might such righteousness look like in practice. Jesus will give particular examples in the succeeding paragraphs of Matthew 5.