According to Peter Guida, the head of NASA's space radiation laboratory, one of the biggest radiation concerns for astronauts (and space-bound tardigrades) is a set of molecules called reactive oxygen species. Ionizing radiation enters the body and bores into wayward molecules that contain oxygen. In simple terms, those newly irradiated molecules then troll through the body causing all sorts of harm.
Tardigrades during their desiccated state produce an abnormal amount of anti-oxidants (yes, these actually exist outside the health-food world), which effectively neutralize those roaming, evil reactive oxygen species. Partly because of this talent, tardigrades have been found to withstand higher radiation doses with far greater success than researchers would otherwise believe they should.