Unless your righteousness exceeds the righteousness of the scribes and Pharisees

The scribes were teachers of the Law and the Pharisees were the most zealous followers of the Law, so by common reckoning and their own generous self-estimation they were the most righteous people around. Jesus, in this passage, is, however, outlining a different, non-legalistic view of righteousness, in which following the rules to the letter is not what gets you into the kingdom of heaven.

It's interesting to note that, even today, so many of us Christians still haven't let go of this old, pre-Christian version of righteousness where it's all about getting the rules right.

The Pharisees and the scribes never kept the word of God. Instead, they obeyed Traditions of the Elders which nullified the word of God.

Jesus Christ severely criticized the Pharisees and the scribes for obeying Traditions of the Elders which nullified the word of God (Matthew 15, Mark 7).

That's why Jesus Christ's biggest opponents were the scribes and the Pharisees (Matthew 23, Luke 11).

Apostle Paul called Traditions of the Elders as "Traditions of our fathers" in Galatians 1:14 and Jewish Priest Josephus calls "Traditions of the Elders" as "Tradition of our forefathers."

"What I would now explain is this, that the Pharisees have delivered to the people a great many observances by succession from their fathers, which are not written in the laws of Moses; and for that reason it is that the Sadducees reject them, and say that we are to esteem those observances to be obligatory which are in the written word, but are not to observe what are derived from the tradition of our forefathers." (Antiquities of Jews Book XIII.X.VI).

That's why you see Jesus Christ saying this.

John 5:46-47 (NIV) - "If you believed Moses, you would believe me, for he wrote about me. But since you do not believe what he wrote, how are you going to believe what I say?”

John 7:19 (NIV) - "Has not Moses given you the law? Yet not one of you keeps the law. Why are you trying to kill me?”

We also see Stephen saying how they didn't obey the laws of God.

Acts 7:52-53 (NIV) - "Was there ever a prophet your ancestors did not persecute? They even killed those who predicted the coming of the Righteous One. And now you have betrayed and murdered him— you who have received the law that was given through angels but have not obeyed it.”

Traditions of the Elders (a.k.a Traditions of our forefathers) existed as early as when John Hyrcanus I ruled (135 BC to 104 BC).

Josephus' Antiquities of Jews XIII.XVI.II - "So she (Salome Alexandra) made Hyrcanus high priest, because he was the elder, but much more because he cared not to meddle with politics, and permitted the Pharisees to do every thing; to whom also she ordered the multitude to be obedient. She also restored again those practices which the Pharisees had introduced, according to the traditions of their forefathers, and which her father-in-law, Hyrcanus, had abrogated."

Traditions of the Elders later came to be known as Talmud. This is agreed by Jewish society.

Rabbi Michael Rodkinson - "The Talmud, then, is the written form of that which, in the time of Jesus, was called the Traditions of the Elders, and to which he makes frequent allusions" (Source - The History of the Talmud, Vol. II, page 70, Chapter IX).

The righteousness of the scribes and Pharisees were extremely low due to the fact that they obeyed Traditions of the Elders which nullified the word of God as I mentioned above. So in order to enter into the kingdom of heaven, your righteousness should exceed the righteousness of scribes and Pharisees by obeying the written word of God and believing in Jesus Christ.

Jesus goes on to show what he means in the following verses.

The Pharisees taught that you should not murder, but Jesus said hating someone is just as bad.

The Pharisees taught that adultery was wrong, but Jesus said lustful thoughts is just as bad.

Many times outside the Sermon on the Mount Jesus corrected the teachings of the Pharisees too. In general, the Pharisees had added a bunch of rules to the law. Their rules were like a fence - if you never crossed their rules you could guarantee you hadn't broken God's laws. But their rules just let them think that they were pleasing God when they weren't. Jesus showed that they need to do everything the Pharisees did too, as well as having a perfectly righteous thought life.

Ultimately Christians believe the standard Jesus set is impossible, which is why Jesus needed to die for us, so that he could take the punishment we deserve and we can receive his righteousness.

The Scribes and the Pharisees were the religious leaders. They were role models for the people to follow. They were blameless in their religious observances. The people looked at them for guidance. The people must have believed that the Scribes and the Pharisees were the first to enter the kingdom of Heaven. However, Jesus was teaching them that God's standard is higher than what man can accomplish. From Matthew 5:21 onward, we clearly see the standard of God for anyone to be counted as perfect. For instance, while the Pharisees never slept with another woman, Jesus was telling them that even a lustful gaze is already an adultery.

We cannot enter heaven with our works and observance to the law, for we can never meet the standard required by God. We need God to save us. Therefore, God gave us His one and only Son, Jesus Christ, a perfect person to take our place and help us reach the standard demanded by God. For Jesus said "No one comes to the Father except through me" (John 14:6).

This is clearly explained by Paul in his Epistle to the Romans.

Romans 3 (NIV)

[10] As it is written: “There is no one righteous, not even one;

[20-31] Therefore no one will be declared righteous in God’s sight by the works of the law; rather, through the law we become conscious of our sin. But now apart from the law the righteousness of God has been made known, to which the Law and the Prophets testify. This righteousness is given through faith in Jesus Christ to all who believe. There is no difference between Jew and Gentile, for all have sinned and fall short of the glory of God, and all are justified freely by his grace through the redemption that came by Christ Jesus. God presented Christ as a sacrifice of atonement, through the shedding of his blood—to be received by faith. He did this to demonstrate his righteousness, because in his forbearance he had left the sins committed beforehand unpunished— he did it to demonstrate his righteousness at the present time, so as to be just and the one who justifies those who have faith in Jesus. Where, then, is boasting? It is excluded. Because of what law? The law that requires works? No, because of the law that requires faith. For we maintain that a person is justified by faith apart from the works of the law. Or is God the God of Jews only? Is he not the God of Gentiles too? Yes, of Gentiles too, since there is only one God, who will justify the circumcised by faith and the uncircumcised through that same faith. Do we, then, nullify the law by this faith? Not at all! Rather, we uphold the law.