。
Is she staying at a hotel?
Literal
She [topic-は] hotel [at-に] is-staying [explanatory-のです] [question-か]?
のですか marks this as an explanatory question — 'is it the case that she's staying at a hotel?' The の hooks the question to a context: maybe the speaker has a reason to wonder (someone said her bag was here, or her schedule looks strange). Without の, the bare 泊まっていますか would just be a flat factual question. 泊まる takes に for the lodging place — に as the location of an action of arrival/dwelling. ~ている here marks the resulting state: she arrived and is currently lodged.