。
She burst into tears at the news.
Literal
She [topic-は] that news [object-を] hearing [then-て] burst-out cried-out.
わっと is a 擬音語 (sound mimetic) for the sudden eruptive sound of bursting into tears or shouting. Combined with 泣き出す ('burst into crying'), the redundancy is intentional — both elements emphasize the suddenness and emotional eruption. 泣き出す itself is a compound verb 泣く ('cry') + ~出す (auxiliary 'burst into doing, suddenly start'). The te-form 聞いて chains hearing to crying, news to tears.