This verse was kind of confusing to me at first, but again, the ESV Study Bible notes helped a lot. Paul says that he died to the law, meaning that he is no longer bound by his obligation to strictly obey the law. This freed him up to live for God—he is no longer bound by legalism, so he is able to serve God joyfully and thankfully. Here's an analogy that applies to my own life: When I am required to practice my trumpet and I have to practice a certain boring song, I don't like to play at all; but when I'm able to play something fun because I want to, and when I'm not bound by time requirements, I actually enjoy playing. Or maybe this applies more to you and makes more sense: I hate reading for school—they just always pick boring books; but I love reading. So if I'm freed from ever needing to read for school again, I can actually enjoy reading on my own time whenever I want. It's the same with the law: Since we are no longer required to obey the law to save ourselves, we can now freely obey God out of thankfullness and love, and we can serve Him joyfully.
What does it mean that Paul died to the law "through the law"? How was he freed from the law by the law? Did the law willingly let him go free?As I write these questions, it's starting to make more sense to me. Because Jesus perfectly obeyed the law and served as a holy sacrifice for our sins, His righteousness freed us from needing to be righteous on our own. Because Jesus fulfilled the law in our name ("Behold, I come in the name of My Father and of My son Michael. I have lived a righteous life, and My Father now deems Michael blameless and holy. His debt has been paid"), we don't have to because we already have by accepting Christ's sacrifice and His forgiveness. Through His obedience to the law, we have been freed from the law so that we can gratefully and joyfully live only to please Him. We no longer have to worry about saving ourselves, so we can now devote all our efforts to pleasing God.