。
She tried not to cry.
Literal
She [topic-は] tears [object-を] shed-not-will tried.
~まい is the negative volitional — a formal/literary way to express 'I will not do X' or 'I shall not do X.' 涙を流すまいとした means 'she tried (とした) to not cry (流すまい).' The ~まい form is relatively rare in modern spoken Japanese, replaced by ~ないようにする in casual speech.