Fig.: things about someone's past that would cause scandal or embarrassment if they were known

He's a very popular senator but he will never be elected president; he has too many skeletons in the closet

È un senatore molto popolare e benvoluto, ma non credo che verrà mai eletto presidente perché ha troppi scheletri nell'armadio

A drug pusher! I can't believe it. She looked such a nice lady, who would have thought that she had such a skeleton in her closet

Una spacciatrice di droga! Non ci posso credere. Chi avrebbe mai detto che una signora così a modo potesse nascondere un tale scheletro nell'armadio

Maybe I'm too cynical, but I think that we all have our skeleton in the closet, and we want to keep it hidden from prying eyes

Forse sono troppo cinico, ma credo che tutti abbiamo il nostro scheletro nell’armadio, e che vogliamo tenerlo nascosto da sguardi indiscreti

This expression refers to an apparently irreprehensible person or family having a guilty secret, and the close-at-hand domestic imagery of a closet gives a sense of the ever-present risk of discovery.
”It has been suggested that the phrase derives from the era of the notorious body snatchers, i.e. prior to 1832, when the UK's Anatomy Act allowed the more extensive use of corpses for medical research. The theory goes that, in a scenario like that of the concealment of Catholic priests in priest holes in domestic houses in Elizabethan England, doctors would conceal in cupboards the illegally held skeletons they used for teaching. There's no evidence at all to corroborate that theory. Concealed skeletons are occasionally found walled-up in houses but they are usually those of unwanted infants” (www.phrases.org.uk)