Edward drives Bella home, and tells her a bit about his life. He was born in 1901, and at 17, he was going to die of Spanish influenza. His doctor, Carlisle Cullen, saved him by turning him into a vampire.
After that, Carlisle also transformed Esme, who had fallen off a cliff and was dying. Esme became his wife. Then he saved Rosalie, who was intended to be Edward's companion, but that didn't work out.
Rosalie found Emmett being attacked by a bear, and carried him to Carlisle to save. Rosalie and Emmett have been together ever since.
Alice and Jasper joined the family later, already as vampires. They had each "developed a conscience" (14.28) on their own and didn't want to prey on humans.
When they arrive at Bella's house, she finds out that Edward isn't unfamiliar with the house – he's been spying on her at night.
[Bella:] "How often did you come here?"
"I come almost every night. [...] You're interesting when you sleep." He spoke matter-of-factly. "You talk." (14.66-67, 69)
To Bella's complete embarrassment, Edward tells her that she says his name at night. "Don't be self-conscious," he says. "If I could dream at all, it would be about you. And I'm not ashamed of it" (14.85).
Bella's dad comes home and Edward quickly disappears. After convincing Charlie that she's tired, Bella heads upstairs and finds Edward in her bedroom.
They snuggle as Bella tries to fall asleep. He feels like it's so much easier being around her now, like he's being desensitized to her smell.
Edward talks about how he's in love for the first time ever. How he's feeling jealousy for the first time (because of Mike), and how love is "more forceful than [he] imagined" (14.184).
[Edward:] "For almost ninety years I've walked among my kind, and yours...all the time thinking I was complete in myself, not realizing what I was seeking. And not finding anything, because you weren't alive yet." (14.200)
Eventually, Edward sings a lullaby and Bella "drift[s] to sleep in his cold arms" (14.292).