“clean out” = to remove things from something
“I’m going to clean out the washer before I put more clothes in. I think someone left paper in some clothing, and it caused a real mess.”

“wash/air your dirty laundry in public” = to talk about personal information in front of other people
“If you want to talk about this matter more, let’s do it at home. You shouldn’t air dirty laundry in public like this.”

“It’ll all come out in the wash” = everything will all be okay in the end
“I know you said some pretty bad things to your husband, but it’ll all come out in the wash. Just apologize for anything you have done, and everything will be okay.