The simplest case - where you only need the Banach fixed point theorem - is quite beautiful if you think about it the right way: your map lands somewhere on the land it marks, so somewhere on the map must be an (extremely small) picture of the map. That picture of the map, in turn, has a picture of the map in it, and this sequence of pictures converges to the fixed point.
(This explanation is from Lawvere and Schanuel's Conceptual Mathematics, which has a great chapter on fixed point theorems where the topology is abstracted and only the categorical content is discussed.)
