The same one that gives them jurisdiction over intra-state drug sales: the "Commerce Clause". The reasoning (which some dispute in the case of drug laws, but let's not get into that here!) is that aviation is inherently an inter-state activity and the subset that's intra-state is too small to deal with separately.

Contrast pilot's and driver's licenses: aside from the history, the reason given for having each state issue the latter with their own standards is that most (but, of course, not all) driving is done within each state. The burdens of having different driving rules between states isn't viewed as great because few people regularly drive in more than a couple of states. But it doesn't make sense to have separate pilot's licenses in every state, even less sense to have different aircraft certification requirements in each state, and isn't practical to have different ATC and airway rules in each state