、。
She nearly cried when she heard the news.
Literal
She [topic-は] that news [object-を] heard-and, on-the-verge-of-crying became.
~そうになる ('come close to / be on the verge of') describes an action that nearly happened but didn't. 泣きそう is the appearance-form of 泣く ('cry') — 'looking like one's about to cry' — and になる adds the inceptive 'come to be that way'. The whole construction lives in that uniquely Japanese aspectual zone of near-actions: things that almost happen, that approach but don't quite occur. Compare 泣きそうだ ('looks about to cry', stative) versus 泣きそうになった (entered the on-the-verge state).