Time travel never makes much sense—ever—but Looper handles it much better than most. The trick, it turns out, is simply to acknowledge that it's crazy, that you'd need to diagram out the forking timelines if you truly wanted to keep track of it all, and even then the paradoxes would overwhelm you if you insisted on thinking about it too hard. Better not to, we're told a few times. Accept the rules that you're given, and go with it. So we do.
And it's worth it because Looper also nails the really crucial element of time-travel storytelling: the emotional logic. The actual mechanics might be nonsense, but the emotional connections between past and present and future ring true and resonate powerfully through all the explosions and gunshots of what is, besides, an exceptionally well made action movie—all the more exceptional for being able to finesse all that goofy time travel stuff.