An exposé is an investigative news article/interview/documentary trying to literally EXPOSE secretive details of a person's life (or, less frequently, details of a private event/ideological movement). Something is damning when it does extreme, irrevocable damage in such a way that mimics the torture of the Christian/Muslim hell.
So a damning exposé is something that a journalist did to uncover details about someone or something, and those details are so grotesque or shocking that the person/thing can never recover from them.
Example: "After the damning exposé on Senator Smith's private life revealed that he'd been frequently soliciting transgendered teenage prostitutes in Thailand for over a decade, he resigned in disgrace and fled the country."