、。
Talking to me like that — who do you think you are?
Literal
I [to-に] such mouth-use-way [object-を] do [quotative-なんて], you [topic-は] what-lord [of-の] intention is [question-だい].
A rich compound sentence with emotional punch. 口の利き方 ('way of speaking, tone') is a set noun phrase literally 'how one uses their mouth.' そんな口の利き方をする = 'to speak in such a way (rude).' なんて flags the preceding clause as something outrageous. 何様のつもりだい = 'what lord do you think you are?' — 何様 ('what great person') is sarcastic/derogatory. だい is a masculine interrogative particle, rough and assertive.