。
The moment she saw me, she burst into laughter.
Literal
She [topic-は] me [object-を] saw [the instant-とたんに] began-laughing.
~たとたんに ('the moment / the instant') is one of several Japanese constructions for tight temporal succession (alongside ~や否や, ~とすぐに, ~なり, ~が早いか). とたんに sits in the middle of the formality range — more vivid than plain ~た時, less literary than ~や否や — and especially common when the second event is unexpected or involuntary. 笑い出す pairs 笑う ('to laugh') with the sudden-onset auxiliary ~出す ('burst into'). The polite ~ました raises this slightly above plain narrative.