Romans 7:7-8 meaning

Paul is writing Romans as a defense of his gospel message against competing Jewish "authorities" who were in Rome at that time. He is likely writing in support of Aquila and Priscilla, fellow Jews from Rome who were co-workers in his ministry to the Gentiles (Acts 18:2; Romans 16:3).

The opposing "authorities" accused Paul of saying that if God's grace increases if we sin, then sinning is a good thing (Romans 3:8). In verse 7, Paul is suggesting that these "authorities" might also accuse him of saying that the law is sin, because Paul just told his audience that sin is aroused by the law (v 5). Therefore Paul asks the rhetorical question (as it would be posed by his opponents):

What shall we say then? Is the Law sin? (v 7).

Paul here is elevating an objection, which he will now answer. If the Law causes us to sin, does that mean the Law is sin? Paul answers adamantly in the negative:

May it never be! On the contrary, I would not have come to know sin except through the Law; for I would not have known about coveting if the Law had not said, "You shall not covet" (v 7).

He quickly explains that the law is not sin (May it never be!), but that it shows us our sin. The problem is not the Law, the problem is our wayward hearts! James makes the same assertion, insisting that we are not tempted by circumstances, but rather by our own "lust" that dwells within us (James 1:14). The Law does us a favor by showing us our current reality, that we are in need of help. But the Law can only show us the problem, it has no power to fix the problem.

Paul points out the Law's benefit of showing us our current reality, saying that he would not have known about coveting if the Law had not said, "You shall not covet" (Deuteronomy 5:12). The command to not covet means it is a sin to desire what doesn't belong to us. If we think objectively about this, once we start wanting "more" we are stuck in a hamster wheel of dissatisfaction—for we can never have "more." Once we get what we desired, it is no longer "more."  Now something else is "more." So we just keep wanting. That means we can never be satisfied. We are choosing a life of constant unhappiness and discontentment.

Paul says he would never have known it was wrong to crave something that belonged to someone else apart from the Law. Paul then explains the general principle that he would not have come to know sin except through the Law. He goes on to explain that the Law gives sin the opportunity to urge us on to transgress, and that sin's desire to break the Law is the reason Paul experiences cravings of every type: Sin, taking opportunity through the commandment, produced in me coveting of every kind (v 8). 

So, without the Law telling us what to do, sin does not have the chance to do something outside of the Law, because apart from the Law sin is dead (v 8). If a child was never told that they could not take cookies from the jar, then taking those cookies would not be wrong. This is why Paul tells his audience that apart from the Law, sin is dead.

Paul is leading to a point where he can show his audience how to choose a mindset/perspective that allows them to live apart from sin, with the reality that we are new creations in Christ. He will pose a mental model of our inner-selves that allows us to separate ourselves from our sinful lusts, and live through the resurrection power of Jesus.