Bella doesn't want to celebrate her birthday because she maintains that she transformed when she was still eighteen.
Alice hands her a key and insists that she go to see the corresponding present. Edward gives her a key of his own, which Bella knows is for her "after car." Her irritation over presents makes Bella realize that, despite her transformation, she still feels like her old human self:
It was a relief and revelation to discover how much of my essential core traits had come with me into this new body. (24.30)
Alice leads Bella and Edward to a tiny stone cottage in the woods, surrounded a small rose garden. It's an image out of a fairytale:
It was a place where anyone could believe magic existed. A place where you just expected Snow White to walk right in with her apple in hand, or a unicorn to stop and nibble at the rosebushes. (24.91)
Alice explains that Esme renovated the place for them to have their own family home and privacy.
Alice leaves Edward and Bella to enjoy their alone time. They both get to indulge in their togetherness without restrictions. When Bella asks Edward if he misses her humanness, he replies that she still feels the same way to him, only he doesn't have to deal with the torture of her human scent.
As night turns to day, Bella still hasn't had enough of Edward. He tells her that none of her desires (for sex as well as the thirst for blood) ever fade; it's a matter of learning how to prioritize them.