、。
When she saw me, she suddenly burst out laughing.
Literal
She [topic-は] me [object-を] saw [and-て], suddenly began-laughing.
笑い出す pairs 笑う ('to laugh') with the sudden-onset auxiliary ~出す ('burst into') — 'to burst out laughing.' Compare with 笑い始める ('start laughing'), which is more neutrally inceptive without the suddenness. ~出す and ~始める are siblings in the family of onset-marking compound-verb auxiliaries, with ~出す reserved for surprise, abruptness, or uncontrolled outbursts. 急に ('suddenly') reinforces the abruptness.