。
Why did she rush out?
Literal
She [topic-は] why rush-and went-out [explanatory-の] is [question-か]?
急いで is the te-form of 急ぐ ('hurry'), here functioning adverbially — 'in a hurry, hurriedly.' 出ていく / 出ていった is a compound 出る ('exit') + いく ('go'), 'go out / leave (toward away from speaker).' のですか combines the explanatory の + polite copula です + question か, giving a polite, slightly searching 'is it that X?' question. This のですか framing is common in polite Japanese for any question seeking explanation; the bare ですか would feel more transactional, while のですか asks for context.