Comments on the individual passages making up Matthew's gospel, chapter 5, can be found here: Matthew 5:1-12, Matthew 5:13-16, Matthew 5:17-20, Matthew 5:21-26, Matthew 5:27-32, Matthew 5:33-37, Matthew 5:38-42, Matthew 5:43-48.
There is a tendency and trap that is easy to fall into which renders the sermon on the mount as merely a quaint lecture giving some good moral principles. The popular notion these days is hence that Jesus was a good moral teacher, who said some important ethical things whilst he was around on Earth, and for those who dislike the Church, that these moral teachings slipped under the radar because we have been far too concerned with theological nuances and fighting heresy. "We want 'blessed are the poor in spirit,' not that dreadful concern for orthodoxy," some say, and perhaps it is true that doctrine has taken up a lot of our time and efforts. It is important, of course, but that is not what I want to discuss presently.
The point remains that to a large extent, it is easy to forget what a startling sermon Jesus is giving. We have seen that the general structure to the antitheses is "You have heard that it was said...but I say to you...," but once we translate the culture into our modern context, it becomes clear how startling that it. Consider if instead it was rendered:
"God had told you...now I say to you..."
This is, after all, exactly what the Jews of the time should have been thinking. This teaching that Jesus refers to merely as something that was said was none other than the words of the Torah, the Law, which God had given to Israel - this was the highest possible revelation, it was considered the word of God! So Jesus is not just saying "there's this moral teaching you've been hearing for a while - that's not quite enough, let me revise it," he is also assuming upon himself the authority to revise or to fulfil the very teaching of God!
In this context, when Jesus says "I have come to fulfil, not to abolish" takes on another meaning: it is also a statement of continuity between the God of Israel and himself. It is a sort of warning against thinking that Jesus is setting himself up and against the "God of Abraham, Isaac and Jacob" - certainly, he presumes the power to give a revised law, but it should not be understood in terms of rupture from the Israelites Mosaic Law.
For these reasons, I think that Jesus is effectively using the "thus says the Lord" that the prophets all used to make clear who they were speaking for. He is saying it, nonetheless, in a way none of the prophets ever did: "Thus says the Lord - me."