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When she heard the news, she burst into tears out of sheer joy.
Literal
She [topic-は] that news [object-を] hear [when-と] joy [from-excess-of-のあまり] burst-out cried-out.
~のあまり ('out of an excess of, from too much X') is a useful pattern for actions caused by overwhelming emotion: 喜びのあまり ('out of joy'), 悲しみのあまり ('from grief'), 驚きのあまり ('from sheer surprise'). The mimetic わっと captures the eruptive sound of bursting into tears. 泣き出す is a compound verb 'cry-burst-out,' from 泣く ('cry') + 出す (auxiliary 'burst into doing'). The conditional ~と after 聞く marks the news-hearing as the trigger that immediately set off the reaction. Crying with joy is itself a culturally recognized response — Japanese has the specific phrase 嬉し涙 ('happy tears').