Hi Anon. Great question. My guess is that Erik saw something like a diamond in the rough in Christine. We know that she had genius, which was apparent while her father was still alive, but his death dimmed her light and only a little flicker of her brilliance was still visible. Erik likely saw this inner fire and realized that Christine had tremendous potential. He perceived the beauty inside of her that few other people appreciated at that time, and he recognized that he could uncover that beauty so the rest of the world could see it, too. Not only that, but he realized that he could tap into her genius, and by melding it with his own, he could create an instrument that would transcend mortal sounds and play divine music.
In a much less altruistic way, I think that Erik was probably also attracted to Christine’s kindness, her purity, and her naiveté. She was sweet and generous to the Opera workers and the little ballet girls, she had a reputation for being virtuous in a profession where it was common for women to become mistresses, and (most importantly) she was a credulous soul who still firmly believed in spirits and angels. This trifecta of Christine being kind, unattached, and gullible made her a perfect target for Erik’s machinations. He recognized that he could win her heart and that he wouldn’t have to contend with another suitor, all while beguiling her into believing that he was a disembodied voice.
As to when Erik actually fell in love with Christine, I think that it was quite late in the novel. He was initially attracted to her, and then became infatuated and obsessed with her, but that is not love. I think that the first time Erik really realized that he loved Christine was when she showed him the meaning of love, and agreed to sacrifice herself to save Raoul, the Daroga, and Erik as well (as I’ve written before, I believe that Erik was trying to provoke Christine into killing him, and everyone else, so that he wouldn’t have to do it himself). The love he felt for her grew when she allowed him to kiss her, and it ultimately overwhelmed him. At that point, he realized that not only did he love her, but he loved her too much to force her to stay with him. It is because of his love for her that he freed Christine, gave Raoul and her his blessing, and allowed them to leave. Christine’s actions fundamentally changed Erik; from her, he learned the true meaning of love and sacrifice, and following her example, he finally did the right thing and let her go, even though he knew it would mean his death.