。
She started crying.
Literal
She [topic-は] began-crying.
泣き始める pairs the masu-stem 泣き with 始める ('begin'), giving a one-word inceptive 'start crying.' Japanese has two main 'start crying' compounds: 泣き始める, which is neutral and just marks initiation, and 泣き出す, which adds a sense of suddenness or breaking out. The choice between them is meaningful: a child who 泣き始めた slowly worked up to it, a child who 泣き出した burst into tears. Both pair productively with most action verbs.