Why would nudists need a washer and dryer ?

To wash towels, blankets & sheets.

For when winter comes or when they go in the civilized world that is different from the "Garden of Eden"?

Washing bedding and other household linen.

havent you seen a naked woman walking around in a loss dress , she has nothing underneath , or a woman with a see thru neglagete .

Presumably they still use sheets, towels, pillowcases and other linens; most 'official' nudists carry a towel with them to sit on--they don't simply put their bare butts on other people's furniture or public seating. 

to wash the brown streaks off the couch cushions

to wash the towels sheets and table clothes

For blankets and towels.

For bedding and towels, I suppose.

Nudists often don't get naked, you might work with some and not know it.