。
She raised her hand because she wanted to ask a question.
Literal
She [topic-は] question [subject-が] wanted-to-do [because-ので] hand [object-を] raised.
質問がしたかった (wanted to ask a question) uses the たい form with が marking the desired action as a kind of subject. ので gives the reason she raised her hand. 手を挙げた uses 挙げる rather than 上げる — both mean 'to raise,' but 挙げる specifically implies raising for attention or voting (挙手する). The distinction is subtle but real.