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She promised she wouldn't go out alone.
Literal
She [topic-は] alone [contrast-は] go-out won't [quotative-と] promised.
ひとりでは layers contrastive は onto the adverbial ひとりで, narrowing the scope: 'as for going out alone (specifically) — she won't.' It implies that going out with someone else is fine; it's the solo case that's excluded. The と before 約束した quotes the content of the promise. 約束する ('to promise') is one of those Sino-Japanese verbs whose noun form 約束 sees heavy independent use as well — promises, appointments, dates, and pledges all live under this single word.