Basically what saber just said. A spring by itself has only potential energy, which is a function of the distance 'x' that it is stretched or compressed. So if you attach a weight to that spring, any potential energy that the spring loses to move that weight becomes the kinetic energy of the weight. If the spring loses some stretch distance 'd' (i.e. rebounds after being stretched), then it lost some potential energy that is easy to find, since it's a function of that distance 'd'. Well, that energy it lost became the kinetic energy of the weight, so clearly that energy must also be a function of the same distance 'd' since the energy is exactly equal.