I can see the first order approximation for a Taylor series: if we want to approximate $f(x)$ near $x_0$, then it's close to the line with slope $f'(x_0)$ that intersects $f(x_0)$, giving
$f(x) \approx f(x_0) + f'(x_0)(x-x_0)$
However, I don't see a good way to understand the higher order terms $f^{(n)}(x_0)/n! (x-x_0)$. I understand the proof of Taylor's theorem, but I don't have any real intuition for it. How were Taylor expansions discovered (i.e. what would lead one to guess the form of Taylor expansions)?