The Pygmalion myth includes the story of Pygmalion, a gifted sculptor from Cyprus, who had no interest in the local women as he found them immoral and frivolous. Instead of wooing women like his friends, Pygmalion concentrated on his art. One day, he discovered a large, flawless piece of ivory and decided to carve a beautiful woman from it. Once the statue was complete, Pygmalion found it so lovely and the image of his ideal woman that he clothed the figure and decorated her in jewels. He named her Galatea, which means "sleeping love". He was so obsessed with Galatea that he visited temple of Aphrodite to ask forgiveness for all the years he had shunned her and beg for a wife who would be as perfect as his statue which the goddess did. The couple spent their lives together and never failed to credit Aphrodite for their happiness bringing gifts to the temple on a regular basis.
Now, understanding the myth as we do, Eliza's bath scene is similar since Eliza is a little like the male Pygmalion. Although she is beautiful, she does not see it. She knows she is a good person, hard working, honest, full of integrity, and dedicated to her art of flower sales. She is not frivolous or wasteful. Rather than spend the money for heat in her little apartment, she sleeps in her clothes and under any extra clothing she may have. She would not in her old life "waste" so much good, hot water on something so frivolous as a bath. Even more, she would never disrobe in front of others--this is where the repeated exclamations of "I'm a good girl, I am!" come from during this scene. Also, like Pygmalion with his lovely statue, Eliza is about to be "awakened" to a side of life she has never known before...and both of them are unable to return to the life they knew before after this experience. Pygmalion falls in love and begs the goddess of love for a wife; Eliza is about to take a similar plunge, and she is unaware of the consequences in full but seems to be aware of the unknown as shown in the bath scene.