。
She studied abroad to relearn her English and polish it up.
Literal
She [topic-は] English [object-を] redo-and polish-add [in order to-ために] study-abroaded.
Two productive set phrases stack here. やり直す is a compound verb (やる + 直す, 'do' + 'fix') — the productive aspectual ~直す is used with many base verbs to mean 'redo / do over.' 磨きをかける is an idiomatic 'polish up / refine,' literally 'apply a polish'; outside language study, it's used for refining anything — a craft, a manuscript, a performance. The te-form yari-naoshite chains the two refinements into a single goal: 'redo and polish,' framed as the purpose of going abroad. 留学する combines 留 ('stay') and 学 ('study') — 'stay-study,' i.e. extended foreign study.