One of the pre-conditions of the theorem is that the domain and codomain are the same. So that's one way to satisfy the theorem. But it's not really intuitive. If you gently swirl a bottle of water, somewhere in the bottle, some water molecule did not move. It's not necessarily in the center of anything. The fixed point could be anywhere. The theorem is about existence only, not construction.
(I am not a professional mathematician, I might also be wrong.)
(I am not a professional mathematician, I might also be wrong.)