The basic structure of it is なんでだ, whose polite form is なぜですか you must be familiar with. The last よ in this case adds the sentence with speaker’s intense emption. The form is interrogative, but actually it just expresses the speaker’s irritation, and blames the listener for it. In English, it’d be like; Who the hell do you think you are!?

A literal translation would be "Why looking down from above behavior!?", where the 上から目線 part is an expression for condescension or arrogance. なんで here is basically "why" - literally it is more like "by what means" but why is the best translation in this case. なんだ is a contraction of なのだ which is hard to translate by itself. It basically takes the preceding phrase, makes it into an adjective and uses it to modify the word "thing" so it would change "arrogance" into "arrogant thing/behavior/action" or it would make "condescention" into "condescending look" or similar. The final よ adds emotional force. That particular sentence your quoting is pretty commonly seen statement in dramatic fiction. A functional colloquial english translation might be "what's with the attitude" or "where do you get off looking down on me".