Eliza has realized that her education from Professor Higgins was really not about social etiquette (speaking properly, dressing nicely, or dancing gracefully) but rather while Higgins was focusing on her exterior, she was learning about how humans truly treat one another and what that says about an individual. Near the end of Act 5, Liza thanks Pickering and tells him:

"I shall always be a flower girl to Professor Higgins, because he always treats me as a flower girl, and always will; but I know I can be a lady to you, because you always treat me as a lady, and always will."

Thus, Liza's words about proper speech and dancing demonstrate her belief that those skills are not enough to give someone self-respect or to make her feel like a lady.  Rather, someone's treatment of another is what makes that person feel like a lady or a maid.  Shaw uses Liza's observation about human nature to satirize the arrogance of his society's upper class and to demonstrate that true "class" cannot be taught.

My understanding on this phrase has always been that Shaw wanted to reiterate his own belief that anybody (and even a nobody) could pretend to be an aristocrat, and maybe even become one. This is a direct dig at the self-centered aristocracy by telling them that they are no different than anyone else.

In other words, Eliza's aim was to get rid of her cockney accent because she felt it was common, as did many others. However, the play on words here given by Shaw was to say that changing her way of talking was no different than changing her dress, style, and mannerisms and yet- she still was the same cockney girl inside.