Jesus' use of the phrase, "You have heard", is accurate since the common people generally didn't have access to a copy of the Old Testament for personal reading; they heard the law read and interpreted by the religious leaders. It should be noted that Jesus does not take away anything from the Old Testament while addressing the topics in this chapter (cf. Matt. 5:18), but He does add His own teaching to it. For instance, Jesus does not deny the truth that physical murder (premeditated killing) is wrong (Exo. 20:13) and that there would be judgment for such (Deut. 16:18). However, He supplemented that teaching by declaring mental murder (i.e., unwarranted anger) to also be wrong and cause for judgment. Jesus used the phrase, "But I say to you", to create a strong contrast between what they had heard of old and His authoritative declarations (cf. Matt. 7:28,29).
The Lord is not doing anything inappropriate here by presenting laws that go beyond explicit Old Testament revelation. Moses announced that the new Lawgiver would speak the words of Jehovah and that His words must be heard and obeyed (Deut. 18:15-20; Acts 3:22,23). Jesus is doing such on this occasion and preparing His disciples for the kingdom of heaven and its righteousness. Even though Jesus' teachings here go beyond the Old Testament revelation, such did not become binding until His death (i.e., when the Old Law was fulfilled; cf. Col. 2:14). Also, nothing that Jesus spoke in this chapter (or anywhere else) contradicted the Old Law in any way. Thus, even if a Jew immediately started living by the principles Jesus taught (i.e., before they were binding), he would not be violating the Old Testament.
In this text, Jesus separates the sin of anger without cause into three categories. Anger that is unwarranted and not righteous exists as: (1) Silent anger--one who is angry with his brother but keeps quiet or to himself; (2) Harsh speech--one whose anger finds expression in labeling someone with harsh words such as "raca", meaning "empty head" or "good for nothing" ; (3) Bitter rebuke--one whose anger finds expression by rebuking someone with bitter words such as "fool". Admittedly, in our culture, this may not seem as bad as calling someone a "good for nothing," but some translators believe that the word used here for "fool" expresses strong feelings of contempt, like wishing the worst upon someone or condemning them verbally. Jesus plainly teaches that to be angry without cause makes one guilty--period. He shows the seriousness of the sin by showing the three respective methods of judgment. First, punishment by the regular Hebrew court, then condemnation by the "council" (i.e., the Sanhedrin or highest Hebrew court), and finally the divine punishment of "hell fire."
Clearly, sin has its stages, and God takes note of it from its inception in the heart. A man's soul is in danger long before his feelings bear the fruits of violence and murder. True righteousness originates from the inside, and Jesus' teachings go beyond the external actions and into the heart (cf. Lev. 19:18). A man will not murder without a motive, and the spirit that excludes hatred and unrighteous anger will make committing the act of murder impossible.
Friends, getting angry is not inherently sinful (cf. Eph. 4:26), nor is the use of the word "fool" (e.g., Matt. 23:17,19). Anger, appropriately expressed in response to rebelliousness against God's will, is righteous indignation. But, anger in response to personal mistreatment is entirely different (and dangerous). Make sure your words are always carefully chosen from a heart filled with love that seeks the best interests of others. If words flow from your mouth that are abusive and injurious, beware--for the fire of hell approaches (James 3:5,6).