。
She worked up the nerve and asked him.
Literal
She [topic-は] mustering-courage him [to-に] asked.
思い切る is a vivid compound verb: 思う ('think') + 切る ('cut') — literally 'to cut off [one's] thoughts,' meaning 'to resolve, to make a decision, to take the plunge.' In its て-form 思い切って it functions adverbially: 'taking the plunge, mustering one's courage.' The image is of cutting through hesitation. たずねる ('to ask') is the more formal counterpart to 聞く. The whole sentence captures the small but real bravery of voicing an uncomfortable question.