“The Righteousness Required”

Jesus came to fulfill the Law and the Prophets not to destroy them. “The Law and the Prophets” can stand for the whole Old Testament. In fact, sometimes the New Testament quotes a verse from the Prophets or Psalms and indicates that it is from the law. So, when Jesus said He came to fulfill the Law and the Prophets, He in essence said He came to fulfill the whole Old Testament. He fulfilled it in that the Old Testament pointed to Him (vs.17).
Jesus did not destroy the Law (OT) in terms of destroying it as the Word of God so that it passed away and is no longer part of the canon of Scripture. In fact, He said not one jot or tittle (the smallest letter or stroke of a letter) will pass away until all is accomplished (vs.18). It still carries God’s authority and must be observed. In fact, one must not break even the least of Old Testament commands (vs.19). The commands have relevance in that
Christ fulfilled them, but their continued relevance for our observance is determined by Christ. He fulfilled the sacrifices with the once for all sacrifice of Himself and so we need not ever offer another animal sacrifice. We are not obligated to observe any of the law of Moses as the law of Moses, but we obey the law as fulfilled and interpreted by Christ. In essence, we are under the law of Christ (see 5:21-48; cp.1Cor.9:19-23).
As true followers of Christ, born of His Spirit, our righteousness exceeds that of the scribes and Pharisees (vs.20), and we will enter the kingdom of heaven (cp. Phi.3:3-9).