In this passage, Paul explains to his readers why and how they are not under the Mosaic Law. Before giving the first reason, Paul gives an example of it: a certain law is binding upon a married woman while her husband is living, but if the husband dies, she is no longer under that law. In the same way, the Romans have died because Christ died, and therefore have died to the Law. Thus, they are no longer joined to the Law, but joined to Christ. In verse 4, Paul states the reason for which they were joined to Christ: to bear fruit, or righteousness, for God. Then, he explains why they must be joined to Christ to bear fruit for God: before they were joined to Christ, they were in the flesh, or evil human nature, and their sinful desires, which were produced by the Law, bore fruit for death, the fruit being sin. However, the Romans have been freed from the Law because they have died to it. Therefore, they serve God in newness of the Holy Spirit, not in oldness of the letter, or the Law.
In verse 7, Paul brings up a possible belief that might be held to by someone who has heard that Christians are free from the Law: the Law is sin. Paul says that one proof that the Law is not sin is that the only way he could have understood sin was through the Law. Then, he gives a specific example: the only reason he could have known about coveting was that the Law commanded him not to covet. Next, he explains how the Law gave him the knowledge of coveting: his sin caused him to covet in every way that he could have coveted because he was commanded not to covet. One reason that this happened is because his sin was dead, or staying the same. In addition, Paul says that, before he learned the Law, he was spiritually alive in his unregenerate opinion. However, when he learned the Law, his sin increased, and he died, or realized that he deserved death. He had believed that the Law was the means by which he would gain spiritual life, but it actually revealed his spiritual death to him. The way that it did this was by causing his sin to deceive him about what would bring him satisfaction. Paul concludes his explanation of the fact that the Law is not sin by saying that it is actually holy, righteous, and good.
Finally, Paul brings up the possible question of whether or not the Law was the cause of his spiritual death. He simply answers by saying that it was not. Rather, sin was the cause of his spiritual death. The purpose for this was that it might be revealed to him as sin, or evil, so that he would see his sin for what it was.