Why, according to Higgins, is Liza an apt pupil in Pygmalion?

In Pygmalion, in Act I, Higgins predicts that, given six months, he'll be able to take Liza "anywhere and pass her off as anyone." He also makes it clear that her accomplishment will be based on her "ear" and "tongue." In fact, he modifies his prediction to say that "if she has a good ear and a quick tongue," his task will take only three months.

The expressions "good ear" and "quick tongue" are representative of the trope metonymy, which is a literary device comprising figures of speech. Metonymy replaces a general concept or an idea with a word representing a concrete object. Therefore a good "ear" refers to the ability to accurately hear various sounds and the fine points that differentiate sounds, while a quick "tongue" refers to an adroit facility in reproducing variations of similar verbal sounds. In other words, possessing a "good ear" and a "quick tongue" means that the individual can recognize and articulate the different pronunciations of a newly introduced language sound system.

In Act III, Higgins tells his mother that Liza gets on "like a house on fire" (really quickly) and attributes this to his earlier predictions being correct: "she has a quick ear." He claims that he will fulfill his prediction (the six month prediction, not the overly enthusiastic three month prediction) and in "six months time" introduce her anywhere and "pass her off as a duchess," thus making him confident that he will win his bet. In summary, Higgins says Liza is an apt pupil because she recognizes audible differences in phonetic sounds and is able to articulate the differences accurately.

In Pygmalion, what quality of Liza fascinates Higgins and Pickering the most?

Higgins and Pickering are fascinated by the way Liza speaks.

Higgins is a scientist of phonetics. When he meets Liza in the rain, he jots down her words phonetically and then guesses her exact birthplace from her accent.

Pickering is a linguist of Indian dialects. Pickering is on his way to introduce himself to Professor Higgins when they meet by accident on the corner in the rain. Pickering observes Higgins recording Liza's words and guessing her origin. He finds Higgins's ability to pinpoint a person's birth place from their accent admirable.

Later, Higgins and Pickering are socializing at Higgins's house when Liza comes and offers to pay a shilling to Higgins to teach her to talk like a lady. Pickering offers to pay for the lessons under the condition that Higgins can pass Liza off as a duchess at an upcoming garden party. Higgins therefore is initially interested in Liza for her cockney accent, while Pickering is interested in whether Liza can change her accent and learn to speak like a duchess.