。
She willingly helped her mother.
Literal
She [topic-は] willingly mother [possessive-の] help [object-を] did.
進んで (すすんで) is the te-form of 進む ('to advance') used adverbially to mean 'willingly, of one's own accord, proactively.' It implies she wasn't asked — she took the initiative. 手伝いをする ('to do helping, to lend a hand') is a set phrase for assisting with tasks, and 母親 (ははおや) is the neutral, descriptive word for 'mother' (as opposed to the intimate お母さん).