The looking-glass is only mentioned once, toward the very end of Act 2. It is involved in what seems to be a very minor incident. Eliza, it seems, has never looked at herself in a mirror, and she doesn't want to start making a habit of it:
LIZA. I tell you, it's easy to clean up here […] Now I know why ladies is so clean. Washing's a treat for them. Wish they saw what it is for the like of me!
HIGGINS. I'm glad the bath-room met with your approval.
LIZA. It didn't: not all of it; and I don't care who hears me say it. Mrs. Pearce knows.
HIGGINS. What was wrong, Mrs. Pearce?
MRS. PEARCE [blandly] Oh, nothing, sir. It doesn't matter.
LIZA. I had a good mind to break it. I didn't know which way to look. But I hung a towel over it, I did.
HIGGINS. Over what?
MRS. PEARCE. Over the looking-glass, sir. (2.303-310)
Given that Pygmalion is itself named after a character from Greek myth, it only seems right to bring up another mythological Greek figure: Narcissus. There's a whole back-story to the thing, but here's all you need to know: Narcissus was a really hot young man. So hot that every girl in town loved him. But Narcissus was vain, and preferred to keep to himself. One day, a god decided to teach the boy a lesson, and led him to a pool of water. When Narcissus saw his own reflection there he fell instantly in love…with himself. Eventually he realized his love could never be, and basically killed himself.
Eliza's own fear of mirrors seems to spring from some fear of vanity. She certainly doesn't want to end up like Narcissus; but it's not clear where her fear comes from. Her father doesn't seem like the type to teach her those kind of life lessons, and we know she never got much help from her mom. Still, she's very protective of her own identity. Whenever anyone questions her motives – like at the beginning of the play, when she's afraid she's going to be arrested – she always pleads, "I'm a good girl!" Wherever her sense of right and wrong came from, it's clear she has one, and she doesn't want to end up like that vain Greek guy.
It helps to begin with the looking-glass not only because it's a single incident that raises all sorts of questions. From here we can talk about another related issue: clothing.