In fact, He takes the Mosaic law and expands upon it. He warns people against not just the action of the crimes of murder and adultery, but the thought of them. His warning is about the damage that our hate and lust can do not just to others, but to ourselves.
Considering how much Jesus seems to impress upon the spirit rather than the letter of the law, it is interesting that He mentions how He is not here to change one letter or even one part of a letter of the law. Today's homily also mentioned the various translations of that line and how they were likely references to specific Hebrew words and letters. Jesus is warning people about looking for loopholes in what God is asking of them. Salvation does not come from a magical formula--instead, it depends on our living our lives continually in the best way that we can.
It is in doing so that we fulfill the law.