The argument is that he had a presence in the US (server) which violated "US's stupid copyright rules". If I was in another country and remote controlled a drone in to the US and blew something up I would be violating US law. Yeah, it's a horrible analogy but it's about as close as you can get where you can visualize the argument.
I don't think it's going to be an easy thing for the prosecutors to prove WHERE he violated US law but that is kind of not an issue any more. KAT is down and essentially gone. The message has been sent -- you dip your toe in the torrent waters above a certain level and your life will be turned upside-down.