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She got into an argument and lost herself in anger.
Literal
She [topic-は] argued-and anger [with-で] self [object-を] forgot.
我を忘れる ('to lose oneself, forget oneself') is a literary set phrase using the classical pronoun 我 ('I, self'), more formal than the modern 自分. The metaphor is of letting one's senses or composure drop entirely — typically due to overwhelming emotion. The pattern 怒りで ('with anger, by anger') uses で as an instrumental/causal: anger as the agent of the loss. 口論する ('to argue verbally') compounds 口 ('mouth') + 論 ('argument') + する.