During an argument with her husband, she got so worked up that she suddenly burst into tears.
Literal
She [topic-は] husband [with-との] argument's occasion, very excited [then-て] suddenly started-crying.
口論 ('argument/quarrel') is a formal Sino-Japanese compound. 興奮して ('getting excited/agitated') in te-form links to the result. 泣きだした uses ~出す to mark the sudden onset of crying.