Deriving Taylor's theorem is not a problem. But I am curious if there is any nice geometric interpretation of the theorem.
I think the answers in the link I added in the comment are really useful. But my own interpretation, possibly not so refined would be two fold.
1) If you consider a "reasonably good-mannered" geometric object, say a curve or a surface, then depending on where you truncate your Taylor series, you can compare it with other well behaved, reference objects locally at least. That is, say for a curve, you can see as to how far it is not a straight line or a parabola/circle and so on.
2) Another possible thread not much different from the first one would be using analyticity, in that the geometric counterparts of such functions(which have a valid Taylor series expansion) are "predictably smooth", in that the smoothness is as you would expect it.
I guess there are a bit too many loose/cheesy words in this answer, but I do hope it helps a little.
You're approximating the given function by a polynomial. The polynomial approximation is constructed so that its derivatives agree with the given function at one particular point.
Geometrically, a Taylor series with two terms is a straight-line approximation; the straight line is the tangent at the given point. One with three terms is a parabola approximation, whose tangent and curvature agree with the given function at the given point. And so on.
Given their myopic focus on a single point, the approximations produced this way are often less than ideal -- you often need an approximation that's good throughout some interval. But at least they're easy to construct.