NewScientist has a new video explaining the relativity twin paradox in one minute.
The paradox is actually due to Paul Langevin in 1911. It should not be so surprising because a similar age difference occurs without relativity. In any realistic space travel scenario beyond our solar system, humans would have to be frozen to withstand the long travel times or the rapid acceleration. Frozen bodies do not age. So a space traveler returning from a long-distance trip would always be much younger than his twin back home.
Relativistic explanations get confusing when you ask whether the acceleration is causing the aging difference. The answer is yes in the sense that the acceleration defines the asymmetry between the twins, and without accelerating one twin back into the frame of the other, you would never notice the aging difference. But time dilation is caused by velocity, not acceleration. Understanding the effect of acceleration on clock synchronization is part of the paradox.