Is it correct to answer "by what?" for the question: "What do you mean?", like:
- What do you mean?
- By what?
- By doing this... (example)
Is there any better way to answer with a question that sounds more fluently?
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Yes, it is appropriate to use "By what?" as a response to the first question. This usage implies that the second speaker is unsure what the first speaker is referring to. I'd say that the interaction you have written sounds fluent in American English.
If you want a more formal way, you can consider using a full sentence: "What do I mean by what?"