Dead Through The Law
The law of Moses (the Torah--Genesis-Deuteronomy) portrayed Jesus Christ as the true Messiah and, therefore, it brought the message of condemnation to Paul, expounding to him his need of another way to justification besides the law itself, which cannot bring justification before God, though it can simply before men. Paul was able to see the perfection of Christ, his fulfilling the law with no guile and without sin, and he recognized the need not only of the law to bring him to Christ, but also the need of Jesus Christ to be his substitute for sin as the law determined and commanded.
The law,then, at one time was Paul's friend, Paul not realizing that his neglect to keep the law perfectly was his sure condemnation to Hell. Then the commandment came, sin revived, and Paul died (to himself and to the law for justification before God, See Romans 7:9). He is dead, then, through the law. Law did its job: to manifest the exceeding sinfulness of sin (Rom. 7:13).
That I Might Live Unto God
If the law could bring life (Gal. 3:21), then justification would be by the law; but since it can bring only death, there is a need for the Spirit of God to bring life into a person, being brought to life by the Spirit by grace: Eph 2:1 And you hath he quickened, who were dead in trespasses and sins; Of course, we know that to be quickened means to be made alive. We know that the letter (of the law) kills, but the Spirit makes alive (2 Cor. 3:6).
That We Might Be Married To Another
Another reason that we are dead to the law is so that we can be married to another, which is Jesus Christ (See Romans 7:4). Through faith in Jesus Christ and the Gospel through grace we are married to Jesus Christ, and we are to be faithful to him as a man is to be with his own wife.
Crucified With Christ
We are crucified with Christ (Gal. 2:20); we have died to the old flesh and are alive unto God (Rom. 6:11). Therefore, it is not we who live (since we are dead), but it is Jesus Christ who lives in us. We live in our flesh by the faith of the Son of God, because he loved us and gave himself for us (Gal. 2:20).
Do Not Frustrate God's Grace
If Paul were to live under the law from which Jesus has set him free, he would frustrate (despise, disannul, bring to nothing) the grace of God; because if works (the law) were added to grace, grace would no more be grace (Rom. 11:6). If grace were added to works, then works would no longer be works. This would bring the grace of God to nothing (make it worthless).
Anyone who believes that he can be justified before God with his works (the law) frustrates the grace of God and depends upon the law under which he is cursed, because no man has kept the law, and has not believed upon Jesus Christ and his Gospel (death, burial, and resurrection of Jesus Christ) for the remission of sins: Acts 10:43 To him give all the prophets witness, that through his name whosoever believeth in him shall receive remission of sins.