The word-for-word literal translation would be "They have washed for each other clothes their". So rearranging the words to make sensible English gives "They have washed each other's clothes" as the correct translation. I hope that helps.
I don't know if you're asking why the word is zao or why it's present at all.
It's literally kind of like "They washed for each other their clothes." I think it could probably also work without zao though, like "They washed for each other the clothes."