。
She went to Austria to study music.
Literal
She [topic-は] music [of-の] study [object-を] do [for-ために] Austria [to-に] went.
音楽の勉強をする packages the activity as a noun phrase ('do music's study'), letting ~ために attach to the full する clause. The shape is interchangeable in meaning with 音楽を勉強する, but the noun-phrase framing reads slightly more deliberate or written. Austria — Vienna especially — is one of the great destinations for Japanese students of classical music, with its conservatory traditions and historic ties to Mozart, Beethoven, Schubert, and the Second Viennese School.