。
She went to Italy to study music.
Literal
She [topic-は] music research [for-のために] Italy [to-へ] went.
音楽研究 — 'music research' or 'music studies' — is a slightly more academic framing than just 音楽の勉強, suggesting formal scholarly work rather than casual study. ~のために attaches to a noun (here 音楽研究) and conveys 'for the sake of / for the purpose of.' Italy, with its centuries of operatic tradition and conservatories like those in Milan and Rome, has long been a destination for Japanese students of classical voice and music history.