The answer to this riddle is "a towel," because a towel gets wetter the more it is used. The solution is based upon wordplay, not the actual physical properties of the object described. A towel gains more water as it dries someone's wet hands or wet body.
Towels can be made of cloth or paper fibers. Paper towels consist of cotton, wood or other plant fibers that absorb water due to the cellulose molecules in plant cells. The sugar inside cellulose clings to water when the liquid hits the dry fibers. Cellulose fibers are not easily split apart once they get into long strings in plant fibers.