The primary motivation is as a technical tool in the construction of Morse functions, to show that one can construct a function whose critical points are at distinct levels. One first constructs a Morse function, then uses gradient-like vector fields to move around the critical points, yielding a different Morse function.
Definition
Given a Morse functionf on a manifold M, a gradient-like vector field X for the function f is, informally:
away from critical points, X points "in the same direction as" the gradient of f, and
near a critical point (in the neighborhood of a critical point), it equals the gradient of f, when f is written in standard form given in the Morse lemmas.