When we hear the phrase of "getting some of one's own" means the same as to get a taste of your own poison, or getting a taste of your own medicine. In Higgins's case, it means to finally come to the realization of how his actions affect Liza and those around her.
Once again, we find a man so selfish and into himself that he still denies that he has caused any damage to Liza by treating her like an object, and not a person. He also uses the excuse that he just treats everyone the same way. However, now he sees the extent of his annoying personality when his mother takes Liza's side and even agrees to hide her from Higgins when Eliza elopes with Freddy. Also, Henry Higgins gets the cold shoulder from Eliza when she thanks Pickering, and not Higgins, for helping her become a lady. For this is memorable to cite what could be the most direct thing said of Henry:
"The difference"between a lady and a flower girl is not how she behaves but how she's treated."
Since Henry really changes but very little, one must wonder what effect Liza's words have in him. We know that he must have repressed his feelings judging by the way that he reacts when she tells him that she needs to go. This is probably the worst punishment: He lost the thing he built and refined. And that thing he could have lived so proud for, left him in return...and thanked someone else.