Relativity implies that a moving clock experiences less passing time than a resting clock (relative to an observer). The "slowness"-factor gets higher the closer you get to light speed, reaching infinity (i.e. impossible) precisely at light speed.

In the example the one twin travels to a distant star with an appreciable fraction of light speed, thus ages slower. Then, the twin travels back, again with an appreciable fraction of light speed, and still ages slower.

This is basic relativity theory. Maybe I understand the article wrong, but from what I read I understood that the twin should somehow "catch up" on his travel back.

I'd love to be corrected.