After Eliza leaves him at the end of act 4, Higgins realizes how much he's come to enjoy her company. Even though the bet is over, Higgins has "grown accustomed" to Eliza's presence in his house and wants her to remain with him. Higgins goes out of his way to try to lure Eliza back: he offers to adopt her as his daughter or for her to marry Colonel Pickering.
Unfortunately for Higgins, Eliza has no intentions of returning to him, because he is unwilling to give her what she truly wants. In the past, she craved Higgins's verbal approval and affection, but he was unwilling to give either. She craves a life with someone who will care for her and decides on Freddy, whom Higgins does not approve of at all. She also threatens to start teaching pupils linguistics, taking all she learned from Higgins and making money off of it herself.
Ironically, while Higgins begins to like this inner strength of Eliza's, he still expects her to run errands for him and return, while Eliza states she will go her own way. Now that Eliza has come to be independent and strong, she wants nothing to do with the domineering likes of Higgins.