I thought it meant that there's a problem now, but a remedy in the future. Think of little kids who get their best clothes dirty at a party, think the clothes are ruined forever, and think they're in trouble for it (parents do make out clothes to be important that way). I know I was told that the stain would come out in the wash. Then (my opinion) the phrase got moved (abducted?) into the realm of metaphor, where something hidden came into public knowledge, thus solving a person's problem, like a false accusation being disproved. (Either I'm following local politics too closely or I need to stop watching those detective movies.)
FWIW, the American Heritage Dictionary of Idioms (1997), explains it as "A problem will be solved or difficulties will disappear." (Didn't I read that in a fortune cookie?) (Getting back to the quote)"Cervantes had the=is metaphoric use of laundry for cleaning up a mess or difficulty in Don Quixote (Todo saldra' en la colada) and it has been repeated ever since."
We warsh in eastern Ohio and western Pennsylvania as well.