。
She volunteered to take on the job.
Literal
She [topic-は] [voluntarily-自発的に] that work [object-を] do-let's [quotative-と] offered.
申し出る is a humble, somewhat formal verb meaning 'to offer, propose, put forward' — a compound of 申す (humble 'say') and 出る ('come out, put out'). The construction [volitional-form]+と+申し出る is the canonical way to say 'volunteered to do X' — the volitional しよう ('let's do, will do') is what's being offered, embedded as a thought through the quotative と. Note the redundant flavor: 自発的に already says 'voluntarily,' but 申し出る itself implies offering on one's own; together they hammer the point.