The phrase "expired in indigent circumstances" is a euphemism for the much harsher phrase "died impoverished." A euphemism is, as defined by Random House Dictionary, "the substitution of a mild, indirect, or vague expression for one thought to be ... harsh, or blunt."
Random House defines indigent as meaning that someone is impoverished and lacking basic necessities, like food and clothing. It's root is the Latin stem word indigent-, which is a verb meaning to lack, to need, or to be poor. The verb expire has one definition meaning "to emit the last breath" as expiration is a technical term in the discussion of respiration. Therefore, if someone is said to have expired, it means they have died.
A possible sentence using the euphemistic expression "expire in indigent circumstances" might be, "The aged man found under the freeway bridge had expired in indigent circumstances." Another might be, "Oscar Wilde, spurned by society after his trial, sadly expired in indigent circumstances in the Hotel d’Alsace in Paris."