。
She got up early to see the sunrise.
Literal
She [topic-は] sunrise [object-を] see [in order to-ために] early-rose.
~ために expresses purpose. 早起きする is a 漢語 noun + する verb — '(do an) early rising,' i.e. get up early. The full proposition 日の出を見る ('see the sunrise') serves as the purpose-clause modified by ために.