It should be easy to see that a non-accelerating object (or person) will trace out a straight worldline. If you ever change your velocity, though, either through smooth acceleration or instantaneous rotation of your velocity vector, you will trace out some non-straight curve in spacetime. If you leave your friend behind and then, at some later time, meet back up again, if you did not undergo exactly the same amount of acceleration throughout your journeys (i.e., because one of you stayed behind and hardly accelerated at all, tracing out a boring straight line path), then you will have different world-line lengths (different "path integrals") between the starting and ending points, and thus will have experienced different amounts of subjective time.
Now, in a Euclidean spacetime, the traveling twin would end up older, because a straight line is the shortest distance between two points. But our spacetime is not Euclidean--it is a Minkowski space, in which acceleration is equivalent to a hyperbolic rather than Euclidean rotation of your velocity vector, so it turns out that straight line is actually the longest distance between any two points, and the twin who leaves and comes back will have a shorter worldline, and thus will have aged less.