What Poland and the Czech Republic will gain by having radars/missiles on their soil, I can't understand.

There's an issue of visas - Poland couldn't get visa-free travel to USA even by sending a lot of troops to Iraq. It's a sensitive issue in both Czech Rep and Poland.

Second, a base always means income (a cynic would say prostitution as well, but hey, who would bother about that in the Czech Republic, sex tourism capital of Central Europe?), and extra income never hurts.

And third, it means that CR and Poland remain glued to the USA even if NATO falls apart (which just might happen: they say any military alliance that had a misfortune of including Hungary had met an untimely death in recent centuries) or the hell freezes over.

So, you see, a lot of advantages. And not too many losses: Russian rockets would fly to the USA over the Arctic, not Warsaw.