。
She struck up a conversation with a stranger.
Literal
She [topic-は] unknown person [with-と] conversation [object-を] started.
見知らぬ — built from 見知る ('to know by sight') plus the classical negative ~ぬ — has settled into modern written Japanese as the default way to say 'unknown / unfamiliar.' In speech you'd more often hear 知らない人. The と here is the comitative ('with'), distinct from quotative と or listing と; the relevant test is that you can paraphrase it with と一緒に without changing the meaning. 話を始める is the everyday way to say 'start a conversation.'