。
She braced herself not to cry.
Literal
She [topic-は] cry-not [intent-と] spirit [object-を] tightened.
Two patterns work together. ~まい is the negative volitional — the negative counterpart of ~よう/~おう ('let's, will'), expressing a determination or intent *not* to do something. 泣くまい reads as 'I will not cry / let me not cry,' carrying force similar to English 'I refuse to cry.' The と after it is the quotative particle, framing the resolution as the content of the bracing action that follows. 気を張る is an idiom for tightening one's nerves, staying alert, or holding oneself together — drawn from a metaphor of pulling 気 ('spirit, energy') taut like a string. The structure 'X-mai to + Y-suru' ('tense oneself with the intent not to X') is a classic Japanese way to render mental resolve.