。
She put on her overcoat before going out.
Literal
She [topic-は] going-out [before-前に] overcoat [object-を] wore.
オーバー is a katakana loanword from English 'overcoat,' shortened in the Japanese borrowing pattern that lops off final syllables (オーバーコート → オーバー). This kind of clipping is common with English borrowings: パソコン (personal computer), アニメ (animation), コンビニ (convenience store). ~前に ('before') attaches to the plain non-past form of a verb to mark the temporal frame: 'before [doing X]' takes the verb in non-past 外出する, not past, regardless of when the action actually happened. The past 着た ('put on') is the past form of 着る ('wear'), specifically the act of donning rather than the state of being dressed (which would be 着ていた).