。
She kept on talking about her family problems.
Literal
She [topic-は] her own family-problems [about-について] [steadily-どんどん] talked-continued.
話しつづける is a compound verb (話す + 続ける) meaning 'to keep talking, to talk continuously.' どんどん is a 擬態語 (state mimetic) for steady, rapid, momentum-building progression — applied to talking, it gives the sense of words tumbling out in unstoppable flow. ~について scopes the talking to the topic of family problems. The image is of someone unburdening themselves at length, perhaps to a captive listener.