I think it’s significant that Jesus probably said these things after the controversies with the Pharisees over the Sabbath had begun (although you wouldn’t guess this by looking at the book of Matthew alone).
Jesus had probably already done multiple healings on the Sabbath earning the criticism of the Pharisees.
But Jesus makes it clear here that,
I have not come to abolish the Law or the Prophets but to fulfill them. (Matthew 5:17)
How did Jesus fulfill the law and the prophets? Primarily in three ways.
First, he fulfilled the prophesies of the Messiah to come.
Second, he was the only man to ever perfectly obey the law. Because of that, he was able to take the punishment for our sin on the cross.
As Paul wrote,
For what the law was powerless to do in that it was weakened by the sinful nature, God did by sending his own Son in the likeness of sinful man to be a sin offering.
And so he condemned sin in sinful man in order that the righteous requirements of the law might be fully met in us, who do not live according to the sinful nature but according to the Spirit. (Romans 8:3-4)
Third, he fulfilled in his death all the sacrifices that God had required of the Jews.
As the writer of Hebrews points out, while God required them, they in themselves could not cleanse us from our sins. What they did do was paint a picture of what Jesus would do for us on the cross.
Now in Jesus, all our sins have been cleansed, and so these sacrifices are no longer required. They are, as Jesus puts it, accomplished in him. (Hebrews 10:1-18)
But while some things in the law were accomplished in Jesus Christ, as was our justification before God, nevertheless, the law still points to the holiness of God and what kind of people God desires us to be.
And so each day, we are to strive to be holy as he is holy. Not in order to earn our salvation, but because now the Holy Spirit lives inside us to help us live it.
Before we were powerless to do what God commanded. But now God himself dwells in us and gives us the power to become the people he originally designed us to be.
And Jesus warns us,
I tell you that unless your righteousness surpasses that of the Pharisees and the teachers of the law, you will certainly not enter the kingdom of heaven. (Matthew 5:20)
That probably stunned the people who heard it because the Pharisees and teachers of the law were considered the strongest preservers of God’s law. They had made rule upon rule in order to make sure they kept God’s law.
But righteousness goes far beyond keeping the letter of the law. Righteousness is a thing of the heart. And if your heart is not right before God, nothing you do is right before Him.
Three points that Jesus would lambast the Pharisees and teachers of the law for were lack of justice, mercy, and faithfulness in their lives. (Matthew 23:23)
We’ll see more of what he means by that as we continue through his sermon.
For now, the question is, how is your heart? You may be doing all the right things, but is your heart right before God? What would he say about your heart today?
Moment to Moment (Bible blog) 