。
She turned down my offer.
Literal
She [topic-は] my offer [object-を] refused.
申し出 ('offer, proposal') is a derived noun — the masu-stem of the compound verb 申し出る ('to formally offer, propose'), itself fusing the humble 申す ('humbly say') with 出る ('come out'). The literal nuance is 'something put forth from one's mouth.' 断る ('to refuse, decline, turn down') is the workhorse refusal verb across casual and formal registers. Compact and direct.