。
She left early to avoid the rush hour.
Literal
[topic-は] rush-hour [object-を] avoid [in-order-to-ために] early departed.
~ために ('in order to') connects a purpose clause to the main action: 避けるために早く出発した ('left early in order to avoid'). ラッシュアワー is the full English-origin compound, more often seen in writing and commuter-information contexts than its clipped counterpart ラッシュ, which is more conversational. The borrowed phrase has settled comfortably into Japanese — fittingly, since Tokyo's rush hour is one of the most extreme commuter phenomena in the world, with packed trains and station 'pushers' (押し屋) historically helping cram passengers in.