[rcxdude]

It's probably most clearly stated in the original post, but the point is you would either be able to find three people who are all friends with the other two, or you would be able to find three friends who are not friends with any of the others. The non-obviousness is due to the fact that someone may have only one friend, or someone may have two friends but those two would not be friends with each other, so I wouldn't call it blindingly obvious that there would always be a subset with one of these properties (I think the main intuition that is useful is that if you have a sparsely connected graph of friends, it's easy to find a group that aren't friends at all, and if you have a densely connected graph then you can find a group that are mutual friends. The theorem is then proving that there's no middle ground: it's either dense enough a mutual group must exist or sparse enough a non-friend group must exist.)