。
She left early to avoid the rush.
Literal
[topic-は] rush [object-を] avoid [in-order-to-ために] early departed.
~ために is the standard 'in order to' construction — verb (dictionary form) + ために. The verb before ために expresses the purpose (here, 避ける 'to avoid'), and the main clause expresses the action taken to achieve it. ラッシュ here is a clipped form of ラッシュアワー ('rush hour'), itself a borrowed phrase. Loanword abbreviation by syllabic clipping is a productive process in Japanese (リモコン from リモートコントロール, パソコン from パーソナルコンピューター).