To talk about your private quarrels or disgraces where others can hear; make public something embarrassing that should be kept secret.
Дословно - проветривать своё грязное бельё на публике. Обсуждать свои личные ссоры или проблемы на публике.
Everyone in the school knew that the superintendent and the principal were angry with each other because they aired their dirty linen in public.
No one knew that the boys' mother was a drug addict, because the family did not wash its dirty linen in public.
She has a habit of airing everyone's dirty linen in public.
У неё привычка обсуждать чужие проблемы на публике.