。
She stepped forward to take his hand.
Literal
She [topic-は] his [genitive-の] hand [object-を] grasp [intending-to-ようと] one-step forward came-out.
~ようと ('with the intention to') is a slightly compressed form of ~ようとする — when used as an adverbial clause modifier ('intending to do X, ...') it can drop する and just attach the rest of the sentence directly. 握ろうと一歩前に出た = 'with the intention to grasp (his hand), stepped one step forward.' 一歩 ('one step') uses the counter 歩 (ぽ/ぽう) for steps. 前に出る ('step forward,' literally 'come out to the front') is the standard idiom for advancing.