Notice in Matthew 5:21-48 how often Jesus says either "it is written" or "you have heard it said" referring back to the Law and/or Jewish tradition. On nearly all these topics Jesus refers to what they have heard or what is written then he expounds upon and sometimes changes these laws with a "but I tell you" (see verses 22, 28, 32, 34, 39, 44). Isn't this rather presumptuous of Jesus to take the Law that God gave to Moses and dare to "reinterpret" it this way? His way if all those "but I tell yous" are a clue as to who is offering opinion on these matters.
I suppose it's no wonder that when Jesus finished this sermon, Matthew 7 records:
28When Jesus had finished saying these things, the crowds were amazed at his teaching, 29because he taught as one who had authority, and not as their teachers of the law.
Someone suggested we should read straight through the prophetic books of the Old Testament (Isaiah, Jeremiah, Amos, Obadiah, Joel, Habakkuk, Micah, etc.) to the end of Matthew 7 in order to better realize the amazement of these people. Whereas the prophets stated, "The Lord God says" in conveying the Almighty's message to the people, Jesus boldly declares, "I say to you." Indeed he taught "as one who had authority"!
Another thing to look for as you read these 28 verses, notice how relationships are important to Jesus. In dealing with hatred towards people, settling matters with an adversary, treatment of women and divorce situations. Even oaths and just letting your "yes" be a "yes" and your "no" a "no." Notice how many of these situations deal with relationships with fellow humans.
I speak a lot about how following Jesus is a relationship and not a religion. Religions are humans' attempts at meeting certain moral standards with hopes that God will have mercy and allow them into heaven. Truly knowing Jesus and following and abiding in him is about relationships. Relationships restored, relationships enjoyed - now and throughout eternity.