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She kept going on about her new car at the party.
Literal
She [topic-は] party [at-で] own [genitive-の] new-car [genitive-の] thing [object-を] talk-keep.
話し続ける is a textbook example of the [verb stem]+続ける compound, expressing 'keep / continue X-ing.' The のこと ('the matter / thing of') interposed before を softens 'talk about her new car' into 'talk about the matter of her new car' — a quintessentially Japanese way of framing a topic of conversation. のことを話す is the standard collocation for 'talk about'; using just 新車を話す would be ungrammatical because 話す ('talk, speak') doesn't take a direct topic that way. 自分の ('one's own') here adds a faintly self-absorbed nuance — the picture is of someone unable to stop bragging or musing about their purchase.