。
She pushed the door shut.
Literal
She [topic-は] door [object-を] push-and closed.
押して閉める chains 押す ('push') with 閉める ('close') via te-form, marking the manner: 'closed by pushing.' Most modern Japanese doors slide rather than swing, but this sentence works for either. The pair of verbs 押す/引く ('push/pull') is the everyday distinction at doors — many Japanese doors carry signs labeled 押 and 引 to clarify direction. 閉める is transitive ('close X'); the intransitive counterpart 閉まる describes a door closing on its own.