Higgins is asking why what becomes of E liza is of importance to anyone, including Eliza herself. It should be obvious to anyone why it is of importance to Eliza herself, and Higgins in asking the question implies that it is of no consequence to anyone. It's just his usual condescending manner.
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probus why what becomes of ElizaThen why his question begins with absolutely irrelevant pronoun 'what'?
dave_anonWhy would you expect that reference to be set off by speech marks?
dave_anon isn't that something you have to pick up by reading the text or seeing the performance?The syntax of the sentence implies that the speaker requests the definition of the collocation 'what becomes of me/you'.
Paul EvdokimovWhat does it matter?This is an idiom for "How does it matter?", "In what way does it matter?" The implied answer is "In no way at all" or "It doesn't matter". Idioms don't always follow the rules of grammar as expected.