The Sermon on the Mount is the greatest sermon ever preached. Jesus often spoke in parables, stories illustrating teachings, but here he speaks quite straight forward, taking the Books of Law (Exodus, Leviticus, Deuteronomy) and interpreting them for what it looks like in the New Kingdom.
Jesus starts his sermon with the beatitudes. Throughout his ministry Jesus often said, "the first will be last," and the beatitudes show this and turn the culture's view of success and contentment upside down. If you were well off in Ancient Jewish culture it meant you had God's favour but Jesus counters this. It's the poor, the mourning, the humble, the hungry, the merciful, the pure, the peacekeepers, and the persecuted who are favoured and blessed by God. Not the rich, powerful, and conquerors.
The section about salt and light is the most metaphorical lesson. Salt was very important in Ancient times because refrigeration was not yet invented. Salt was need to keep food from going bad as well as a seasoning to make food taste better (as it still does today). Jesus is calling the people to preserve the goodness of the world which has fallen into sin. Light helps us see what is around us. Jesus is calling the people to show people the Truth, preparing people to preach the Gospel.
Next Jesus talks about the law. He doesn't want the people misunderstanding what he is saying as he is about to go about talking about the law. He isn't there to get rid of the Law of God. The commandments of God are the foundation to how we are to interact with each other and with God himself. Some of the law no longer applies because it is things to do with our self (what we wear and what we eat) and some of the things no longer apply because Jesus has made us righteous enough to stand in God's presence (Sacrificial rituals) but things like the Ten Commandments will never become something we can ignore.
Not only does Jesus say he's not there to abolish the law but he actually raises the bar. The law says if you commit murder you have committed a sin, but Jesus says if you even think about committing murder you have sinned. Jesus has come not to just deal with the outward actions of sin but to heal our hearts, which since the fall have always been, "consistently and totally evil" (Genesis 6:5). Jesus deals with the root of the probelm, our separation from God has made us evil, by sacrificing himself Jesus brings us back to God who can heal our hearts to their original creation, good.