。
She started crying when she saw my face.
Literal
She [topic-は] my face [object-を] saw-and cried-began.
泣き始める is the conventional pairing for 'begin to cry,' built from 泣く (cry) + ~始める (begin to). The auxiliary ~始める marks neutral onset — and stands in contrast to ~出す (sudden, often emotional eruption, as in 泣き出す 'burst into tears'): same image, different intensity. The te-form 見て here links cause and reaction — 'saw, and then began to cry' — leaving the emotional logic implicit.