。
She turned down my offer to help.
Literal
She [topic-は] aid let's-do [quotative-という] my offer [object-を] refused.
援助しようという is a tidy stack: the volitional 援助しよう ('let's help / I shall help') becomes the content of 申し出 ('offer') by way of という, the quotative connector. The construction lets the offer be defined by its proposed action — 'an offer to-the-effect-of "let me help."' This volitional + という pattern shows up wherever proposed or intended actions need to attach to abstract nouns: 行こうという計画 ('a plan to go'), 助けようという気持ち ('a feeling of wanting to help'). 申し出 is itself a noun derived from the compound verb 申し出る ('to offer / propose,' literally 'say-out'), and 申す is the humble form of 言う — so 申し出 carries a slight sense of formal proposal.