Proof (preferably related to geometrical intuition) of the validity of Taylor Series?

I understand that a funtion can be approxiamted by the following equation:

But I don't understand why??? I mean, is there some explaination (proof) that involves geometrically (as in something that uses a x-y graph) intuitive concepts? And if there is no such proof, can somebody provide a proof of the fact that the above equation (when the sum is repeated indefinantly) is infact equal to the function in question? And I'd prefer a simpler (in terms of mathematical knowledge required to understand it) proof if possible. But, if there is no way around the complexity of the proof, can somebody at least explain parts of the proof in their post? Thanks in advance