To summarise from Wikipedia, the story of My Fair Lady is as follows:
The story concerns Eliza Doolittle, a Cockney flower girl who takes
speech lessons from professor Henry Higgins, a phoneticist, so that
she may pass as a lady.
In other words, taking a common girl and making her into a proper lady.
The lines you refer to are the professor telling her that a proper lady would wipe her face with a handkerchief, not her sleeve. Only a common girl would use her sleeve. Thus, if she used her sleeve it would give away that she was from a common background - which would be bad, since Henry wants her to pass as a lady.