Covenant means "to cut." It was a way in ancient times that people would exchange promises to each other. You would "cut a covenant," splitting an animal into two pieces. This animal was usually provided by the "inferior" party. The split animal was to symbolize "let it be for us like this animal if we should break these promises."
God made numerous covenants with man. He made a covenant with Abraham that through him all the nations (peoples) would be bless. He made a covenant with Israel (through Moses) providing them the law, in which if they obeyed, they would live, but if they disobeyed, it would bring forth death. There was also the covenant with David that through his bloodline a King would reign over Israel forever. Some of the covenants had conditions and some didn't. If they did have conditions, Israel never failed in breaking them.
When you read "the law" in scripture, you can think covenant. So when Jesus says "Do not think I have come to abolish the law or the prophets. I have not come to abolish them but fulfill them," you can read it as "Do not think I have come to abolish the covenant or what was foretold. I have not come to abolish them but fulfill them."
This verse is often used to buttress different forms of Christian legalism. However, in it's context this passage destroys all forms of legalism, and my explanation will address the verses that follow it.
On the cross, Jesus fulfilled the covenant. He not only fulfilled the covenant in his life, but he fulfilled the covenant in his death by becoming that cut "animal." Israel had been in a perceptual cycle of animal sacrifices to reestablish the covenant because of their unfaithfulness. On the cross, God provides and shows himself once again faithful.
Jesus also does something else. He not only fulfills the terms of the old covenants, but he cuts a new covenant in his own blood. This new covenant combines all the previous ones, and this time it will not between God and an unfaithful Israel. It will be between God and a faithful Israel who is Christ.
The message throughout scripture is of God's faithfulness and provision.
So what about the rest of the passage I quoted earlier?
At first glance this seems daunting, especially to the audience that first heard it. To us, scribes and Pharisees is synonymous with self-righteous hypocrites. That wasn't the case in that time. In fact they were highly respected. Try to think of a pastor/priest combined with a public official, and you might get the idea of how Pharisees were viewed. If I'm a first century Jew hearing this, my heart is sinking. Exceed their righteousness? They're the best of us.
However, those who are in Christ have his righteousness. In Paul's letter to the Romans:
"But now apart from the Law the righteousness of God has been manifested, being witnessed by the Law and the Prophets, even the righteousness of God through faith in Jesus Christ for all those who believe; for there is no distinction; for all have sinned and fall short of the glory of God, being justified as a gift by His grace through the redemption which is in Christ Jesus"While some might disagree with me, it is my opinion that God never needed sacrifices to forgive. I believe that is purely a pagan concept of what the gods were like. The gods were angry, and they needed sacrifices to be appeased. To show how absurd this idea is in regards to God, the ancient people believed that the gods would come down into the fire while the sacrifice was being burned and eat the sacrifice. We know that God isn't in need of food, but we see him playing along, even saying things such as "it is a pleasing aroma to me."
What I believe is this was God, meeting people where they were at, taking on the appearance of what they understood the gods to be like in their culture, and adapting it to point them to Christ.
I don't believe that Jesus died so that God could forgive. I believe the cross shows us that God does forgive, and if we demand blood (as we so often do), then he will provide his own blood, and he take that hate and malice turn it into forgiveness. That love and forgiveness has the power to cleanse and transform us.