It´s easy. They don´t!

Why? Water shortage, no washing machines, too expensive transportation from Earth…
But, don´t worry.. They are not your smelly buddy in one piece of clothing for the whole time they are there. Thanks to the carefully controlled environment, their clothes don´t pick up dirt from outside, and they don´t tend to sweat a lot.. When they get too dirty they just put the cheap stuff in cargo spacecrafts which burn up in Earth´s atmosphere. More expensive items, like spacesuits are sent back to Earth for cleaning and repairs.

We found, that Tide has developed a fully degradable detergent, specifically designed for use in space. So maybe there will be soon a laundry shop in space too…. Just kidding