。
She turned her face away and started crying.
Literal
She [topic-は] face [object-を] turning-away cry-began.
顔をそむける ('to avert one's face') sets up the body-language frame: she breaks eye contact first, then the tears come. The te-form 顔をそむけて links the two actions in tight sequence — first action flowing into the second. The compound 泣き始める attaches the inceptive auxiliary ~始める to the verb-stem of 泣く, marking the start of the action. Compare 泣き出す, which would imply a more abrupt outburst — 泣き始める is gentler, the tears building rather than erupting. The whole sentence is a small cinematic gesture of someone trying not to be seen crying.