Integrals of the form occurring here are called line or path integrals over the line L. This equation follows from the same result in one dimensional calculus when L is a straight line. But it holds just as well when L is a sufficiently simple and smooth path in higher dimensional space. Such a path can be broken up into tiny pieces, each one arbitrarily close to a straight line, so that on each one the statement holds to arbitrary accuracy. Adding these statements up implies the claim for the entire path.
This statement implies that the line integral of a gradient depends only on the endpoints of integration and not on the details of the path chosen to get from one to the other.