Bathing in Space works pretty much the same way a hand sanitizer, dry shampoo works on the earth. When in places where water is not readily available, once can thank technology for such solutions. The difference is never to spray them in Space. Its bathing time, yay!
The cleansing in space takes place in a similar way. The soaps or shampoos are applied to a slightly soaked area, enough that it gets absorbed into the skin and doesn’t float away, and then the soaps and shampoos are applied. A small amount of water is used so that it can be absorbed in skin and help avoid any escapes. The water sticks to the skin due to Surface Tension, and no-rinse soaps or shampoos help you bathe. Simple rubbing or massaging motion will clean it right away. In case of hair, the wet hair is massaged with a no-rinse shampoo that is later dried off with a towel.
Samantha Cristoforetti, an Italian astronaut in European Space Agency shows you how.