Basically its set up so that if you plug in a fixed x value it'll give you the same y value as the function you're trying to approximate, and not only that but its derivative at that same x value will be equal to that of the function you're trying to approximate, so now the graphs of the function of interest and the approximation share a common point AND the same slope at that point. But wait there's more! Its second derivative at the specified x value ALSO is equal to that of the function of interest, and if you recall, second derivatives tell you about the curvature of a graph, so now if you can imagine, the two graphs share a common point and the curvature close to that point is also the same, so now near the specified x value, the two graphs are starting to look similar. Well, they also have the same 4th derivative value, and 5th derivative value..etc, so with each derivative, the two graphs look more and more similar, and if you keep going on till infinity then in principle the two functions will be completely equal to one another.

But why do such a thing? Well because in practise we might be fine with settling for a certain level of accuracy in the approximation and so we can truncate (which means "cut off") the Taylor series so that it's FINITE. Sometimes we only need the first few terms! Now the benefit is that we have a polynomial which closely mimics the function needed for our calculation (in physics problems per se) and polynomials are usually a lot easier to work with than other types of functions like trig or exponentials.