This theory states, basically, there is at least one point on that function that sends the point to itself. That is, there is always at least one point x such that, for a function f, f(x) = x.
I find this property of life so unequivocally awesome that I have to talk about it.
What this theorem shows is that regardless of how you manipulate the world around you, something will end up back where it started.
Let's start simple...
1. Imagine you had a flat map of your country sitting in front of you...or, indeed, dear reader, go get one now. Were you to drop it on the ground, there would be, without a doubt, one point on that map that was directly above the point it represented no matter the scale of the map as long as one sat inside the other.
2. Alternatively, take two pieces of paper with identical pictures on them, crumble one up and place it on top of the other piece of paper. At least one point on the crumbled up piece of paper would be sitting above its corresponding point.
3. Let's say you had a cup of coffee and a spoon. After stirring your coffee, there would always be at least ONE atom of coffee in that cup that was in the exact same place as where it began.
4. This property is also the reason why you could never have a tie in a game of Hex.
As for the proof itself, it's
The proof is normally done by contradiction--that is, we try to prove the opposite (i.e. that there are no fixed points) in the hopes that we find a result that shouldn't exist.
Effectively, we begin by saying "Assume there is no fixed point" and then try to "break" the proof using what we already know. If we can show that saying there is no fixed point is absurd, then there can only be one recourse...that there is at least one fixed point somewhere.
Interestingly enough, later Brouwer rejected this proof because he felt that all proofs must be "constructed." That is to say, he felt that proof by contradiction was cheating and that if you are going to prove something, you should do it directly--which he did end up doing for some proofs, including, eventually, the Fixed Point Theorem**.
I hope you feel more enlightened now.
*I know the feeling quite well. My first time being given this proof was in a special guest lecture about the history of proof. It whizzed by me so fast that when I copied it down, even now it makes no sense...which is a terrible shame.
**Which was, I'm sure, awesome for him. Even I don't want to touch that one.