。
She started crying loudly.
Literal
She [topic-は] loud-voice [with-で] crying began.
泣き始める is a compound verb 泣く ('cry') + 始める ('begin'); the productive ~始める pattern means 'begin doing X.' The first verb appears in its ます-stem, the second carries inflection. 始めました is the polite past of 始める. Compare 泣き出す ('burst into tears'), where ~出す is more sudden, vs 泣き始める, which is more neutral about the onset.