As with any teacher-student relationship, it's best if there's a firm break between being in that relationship and being in a relationship of equals.
That's what this scene is. Eliza is now fluent in upper-crust English. She can "Stand on [her] own without [him]." So, what she's saying is that she doesn't need him as a teacher any more. And, based on his reaction, we are led to hope that he no longer sees her as a student.
In the time between then and the end of the film, when she's hanging out with Freddy, she realizes that, while she may not need Henry as a teacher any more, she needs him as a man - she needs a partner. She's fallen in love with him. And, based on I've Grown Accustomed to Her Face, he has similarly fallen in love with her.
Freddy is offering to love her but not in the way she wants - his love is childish, it's all about making pretty speeches. This is pretty well discussed in Show Me. She wants someone who does things that show he cares rather than flowery language. Henry doesn't do flowery language but he's done everything else for her. He's given her a better life, made her self-sufficient.
So, when she returns to him, she returns as a woman who loves a man, not as his student and he is a man who loves her, in return.
As to why she doesn't require an apology - first, I don't think she'd expect one from him. That's not the sort of man he is and she's already said that she's tired of words. Secondly, I'd say she actually gets one, of a sort. There's an action in that scene that more than makes up his "apology". He enters the room and plays a record of her voice from when she first arrived. This confirms to her that he loves and misses her in a way that's much better than any spoken apology ever could.