" I wager most Americans have no idea of what is going on in the rest of the world"
Most of the British have no idea what's going on in Africa either.
BBC news inside Britain has very little on Africa, unless there is a new famine or war to report on.
On average, I'd say Americans probably know about Asia than the British do.
This of course only reinforces others belief that we don't care about the rest of the world. Who then treat us with a condescension that is humiliating when it isn't merely annoying. I've had foreigners assume I'll not be able to pronounce the name of the country they come from, (once an eastern European I met was shocked that he didn't have to explain where Moldova was to me). Needless to say I was miffed at his surprise.
So of course this attitude about Americans just gets our hackles up and tends to make one act like we actually don't care. I think those who are the most hurt by this attitude even believe they have stopped caring.
But I'm only hoping this is the perception of your average Joe Six-Pack or Pierre Cheeseater. Add a healthy dose of revolutionary agitator's rhetoric in this or that foreign media, and, not a small amount of a superiority complex and you get modern Anti-Americanism, and the American reaction which feed each other in a reciprocal relationship. What with the populism which drives and empowers the respective governments, it must be a diplomatic nightmare. Nothing like the almost genteel regard for one's enemies, let alone allies, which was common enough 200 years ago.