I (sort of) understand what Taylor series do, they approximate a function that is infinitely differentiable. Well, first of all, what does infinitely differentiable mean? Does it mean that the function has no point where the derivative is constant? Can someone intuitively explain that to me?

Anyway, so the function is infinitely differentiable, and the Taylor polynomial keeps adding terms which make the polynomial = to the function at some point, and then the derivative of the polynomial = to the derivative of the function at some point, and the second derivative, and so on.

Why does making the derivative, second derivative ... infinite derivative, of a polynomial and a function equal at some point ensure that the polynomial will match the function exactly?