Paul's argument in the beginning of Rom 7 always fascinates me:
Do you not know, brothers and sisters—for I am speaking to those who know the law—that the law is binding on a person only during that person’s lifetime? Thus a married woman is bound by the law to her husband as long as he lives; but if her husband dies, she is discharged from the law concerning the husband. Accordingly, she will be called an adulteress if she lives with another man while her husband is alive. But if her husband dies, she is free from that law, and if she marries another man, she is not an adulteress.
In the same way, my friends, you have died to the law through the body of Christ, so that you may belong to another, to him who has been raised from the dead in order that we may bear fruit for God (Rom 7.1-4).
The argument is a simple one but at the same time massively profound. Effectively the idea is that a person is bound by the law only so long as that person is alive at the same time as the law. The Roman audience of Paul's letter, however, has died to the Law in some mysterious way through the body of Christ. Consequently they are no longer bound by it.
Presupposed in this argument is a robust doctrine of union with Christ. In some significant and powerful way, the Christians must be actually united with Christ such that his death was actually theirs as well. Therefore just as Christ died towards the law, so they too are dead towards the law. But also presupposed is this: Christ's resurrection in some way puts him beyond the domain of the law. Otherwise his resurrection would merely be a return to the law's domain. Resurrected life must be of a different category, such that the law no longer has any authority over it.
How do we get united to Christ in this way? Arguably the answer Paul gives is: faith and baptism. In fact the two are connected so tightly in his mind that they are treated as if they are simultaneous events. It is obvious that "faith" is an important presupposition of salvation for Paul. Again and again he emphasizes that salvation comes to those who believe the gospel and who exercise faith. Yet it would equally seem that baptism is a condition of union with Christ, since he affirms that we are united to Christ through means of baptism.
He says that all of us who have been baptized into Christ Jesus have been baptized into his death (Rom 6.3). Likewise he says that in Christ Jesus you are all children of God through faith, but that As many as you as were baptized into Christ have clothed yourself with Christ (Gal 3.26-7). Baptism is clothing yourself with Christ: it is uniting with Christ so that, among other things, you have died to the law through his body along with him.
To address now the second point, Paul's argument likewise presupposes that Christ's resurrected state is in some way beyond the domain of the law. As I've said, if the law were universally binding across all times for the rest of history, Christ's resurrection would simply return him to the rule of the law, and there could be no argument from our death in him to our freedom from the law. But how is this to be understood?
I think the implication is that the law was conceived as something characterized this age and this life, and not the life in the next age. We might even suppose that the law was understood to be a means of acquiring life in the next age, even while the law itself is left behind. Jon D. Levenson describes rabbinic interpretations of the law along these lines in Creation and the Persistence of Evil (Princeton, NJ: Princeton University Press, 1987), p. 35.
But if the law is only a means to acquiring life in the next age, then those who already enjoy that life have no need of the law. Consequently they are not under the law. This, I think, is Paul's understanding and argument. If we are united to Christ and enjoy all that this entails -- righteousness, the Holy Spirit, etc. -- then we have no need for the law. We got what it was for without using it.