That is a wonderful question, because it touches on the central idea of the play: Does society mold you, or do you mold yourself?
In the case of Eliza, we should take into consideration that she already had an inkling that she wanted a change for herself. Like she says in Act 2,
THE FLOWER GIRL:I want to be a lady in a flower shop stead of selling at the corner of Tottenham Court Road. But they won't take me unless I can talk more genteel. He said he could teach me. Well, here I am ready to pay him—not asking any favor—and he treats me as if I was dirt.
This being said, we can safely assume that Eliza already wanted to be someone different; the "call" was already within her. The idea about the bet was the creation of Higgins, alone to boost his own ego. Therefore, Eliza's change in mannerism and speech is mostly the icing on the cake; she already felt the need to become a better person.
Yet, not all is as rosy as she pictures it, because when she learns to speak properly she...
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