Its not at all a silly question if you've never been there.
Things vary a bit depending on the size of the airport, but airports of all sizes in the US are typically government owned. There is usually public transient parking at no charge if you leave before nightfall. If you stay overnight, there is usually a small per night charge for the transient ramp.
FBOs are businesses that lease a plot of land from the government to sell fuel etc elsewhere on the airport property. FBOs vary in size and scope greatly, from shacks to Taj Mahals. The shack type might not be manned so your buy your fuel from a self serve pump. If nobody is around and parking arrangements aren't obvious you use good judgement and put it someplace where it probably won't get in the way. If somebody happens to be around when you return, you ask them nicely if you owe them anything and very often you don't.
On the other end of the spectrum are the FBOs where you taxi up to the fuel pumps under a jet sized awning, get out without exposure to sun or rain, and walk away to your rental car. They fuel the plane, park it where they want it, and you pay them when you get back. Prices vary, and at those kinds of places they have people in shorts and golf shirts to answer your questions.
And of course there is everything in between.