It's one of the moments where the film transcends its origins on stage and gives us a perspective that's impossible in theater. There are others; you won't find any montages of Eliza learning to speak in the original play (and I'm not sure what a montage would look like on stage, obviously). When adapting a play for film, there isn't any reason to shoehorn in tricks that only work on film, unless, and this is key, they allow you to come closer to the truth of the characters or situations than you could on stage. This tracking shot is a classic example, taking us from the dark, muddy streets to the pinnacle of British society in a single turn of the camera. It's just as dazzling for the audience as it is for Eliza. And yet the camera is following Karpathy, the phoniest character in the entire film. Well, Scorsese ended the Copacabana shot with Henny Youngman telling terrible jokes, so I guess it's a tradition.
The only thing that matters in an adaptation of Pygmalion is the chemistry between Higgins and Eliza. As you probably remember, the two fall madly in love over the course of her tutelage. But you're not remembering Pygmalion, you're remembering My Fair Lady. According to his own introduction to the play, Shaw wrote Pygmalion to bring attention to the importance of phonics. I'm going to say that again: Shaw thought, or claimed he thought, he was writing about phonics. Here he is in his own words:
The English have no respect for their language, and will not teach their children to speak it. They spell it so abominably that no man can teach himself what it sounds like. It is impossible for an Englishman to open his mouth without making some other Englishman hate or despise him. German and Spanish are accessible to foreigners: English is not accessible even to Englishmen. The reformer England needs today is an energetic phonetic enthusiast: that is why I have made such a one the hero of a popular play.
In fact, Shaw was opposed to the idea that Higgins and Doolittle should fall for each other, saying in an interview, "I cannot conceive a less happy ending to the story of Pygmalion than a love affair between the middle-aged, middle-class professor, a confirmed old bachelor with a mother-fixation, and a flower girl of 18." When you put it that way, he has a point.