。
She went to great pains to become able to speak English.
Literal
She [topic-は] English [object-を] can-speak [become-ようになる][quotative-と]-tried, bone broke.
Three layers stack to express the goal. 話せる is the potential of 話す ('to speak' → 'be able to speak'). ~ようになる takes a verb and adds 'come to do / reach a state of doing,' so 話せるようになる = 'come to be able to speak.' Its volitional ようになろう ('let me reach the state where I can speak') plus と marks intent — a deliberate aim — and 骨を折る ('to break one's bones') is the body-part idiom for taking great pains, working hard at something. The whole construction reads as 'she put in effort with the intent of becoming able to speak English.'