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The instant she stepped into her room, she burst into tears.
Literal
She [topic-は] her own room [into-に] entered [the instant-とたん] cried-burst.
~たとたん attaches to a past-tense verb to mean 'the very instant X happened, Y followed immediately,' implying almost no gap between the two events. Without に here (compare ~たとたんに), it reads slightly more colloquial. 泣き出す is a compound verb (泣く 'cry' + 出す 'put out / start), where ~出す attached to verbs of action marks the sudden, often involuntary onset of an action. Combined: the moment she got inside, the tears burst out — a stock pattern for the breaking of held-back emotion.