In biblical times salt was used as a means of preservation, and Jesus links Christians and salt together by that means. He tells the Christians gathered that they are the salt of the earth, but also warns them not to lose saltiness. Technically speaking, if the salt molecule bonds with dirt, it loses its potency and will not have the same preserving qualities that it had before. In the same way, Christians will become ineffective if they are bonded by the sins and temptations of this world.
In verse 16, it is interesting to note that Jesus tells the people to let their light shine before men, that others may see their good deeds and praise the Father in heaven, not the person doing the good deeds. So often we either do things publicly in order to draw attention to ourselves, or we shirk from that good doing in fear that we would be considered proud in doing it. Both ways are wrong. We should do good deeds in a way that we point to the Father, not to ourselves.
Apparently many people had been saying that Jesus was going against what the prophets taught in the Old Testament. However, he remind them that he didn't come to abolish them, but to fulfill them. He then cautions people that anyone who breaks even one of those laws or teaches others to break them is not fit for heaven. In order to go to heaven by themselves, they would have to be completely perfect. This was certainly a foreshadowing of Jesus' purpose on the cross.
The Pharisees and the Scribes had limited murder to the physical act and had perverted the commandments of God by making them into something that was simply outward. To the religious teachers of the day it didn't really matter what your heart was like as long as you looked right on the outside. Jesus contrasts the views of man with the views of God of murder. The outlook of the Pharisees and the Scribes disregarded the offense of murder.
As long as you didn't physically kill someone, your heart could be filled with whatever murderous thoughts you wanted. This was against the Old Testament law however. Leviticus 19:17 "You shall not hate your countryman in your heart."
However, the law requires more than outward appearance. Jesus tells the people in Matthew 5:20 that "unless your righteousness surpasses that of the Pharisees, you will not enter the kingdom of heaven." The righteousness that is required is far more than just the outward obeying of the commandments. The requirement to get into the kingdom is an inward obeying as well. Thus Jesus breaks down the Rabbinical teachings and also the illusion of self-righteousness.
It must have stunned the people to hear Jesus tell them that anyone who hates or slanders his brother is guilty of murder, because not actually committing the act of murder is possible to do, but to never hate someone was impossible. Verse 22 says, "Anyone who is angry at his brother is liable to the courts." This anger is not righteous anger like that of Jesus in the temple or the anger of Moses at the golden calf; it is alright to be angry when others are being abused or the God's Word is being mistreated, but hate due to selfish reasons is sin.
In order to not harbor this hate, Jesus tells the people to settle matters quickly with whomever you are angry. Not only will this be good for the relationship between the two people, but more practically Jesus tells us that we won't get in as much trouble with the law if we really do commit the murder.
If someone was sacrificing an animal on the altar, supposed to be one of the most sacred and dignified things, he is to leave it and go be reconciled, then offer the gift when he is finished. Jesus showed the people that harboring hate while giving God an offering is like offering him rancid meat; how much better to give God a gift that is pure and spotless!
