。
She started crying when she saw his face.
Literal
She [topic-は] his [genitive-の] face [object-を] saw-and started-crying.
泣き始める ('start crying') is a 複合動詞 from 泣く ('cry') + 始める ('begin') — a productive pattern where the auxiliary 始める marks the inception of any action. Compare with 泣き出す ('burst into tears'), where 出す marks a more sudden onset. The te-form 見て here links cause/sequence ('saw and so').