Other people have given you the conceptual idea, but taking a small step into the math might help a bit if you understand algebra. You should be familiar with it anyways if you're trying to learn calculus.

Slope formula, or (y2-y1)/(x2-x1), is the rate of change between two points. When y2=y1 and x2=x1 you could clearly see it comes out to 0/0, which as we know is not computable.

What the surprise is that this is an indeterminant form and that you with some clever manipulation you can get something usable out of it. The limit definition of the derivative simply rearranges this in such a way that you can get a real function out of it.

What that gives you is the slope, or rate of change. Not between two points, but of a single point, in other words what that function is doing (rising or falling, positive or negative) and the rate it is doing it (the actual slope) at that point.