In New Moon, Edward lies and leaves Bella because after the incidents at her eighteenth birthday party, he realizes that, even after toiling so long to protect her, she isn't safe from anyone, including him and his family.  He lies to her when he leaves in the hope that she will think that he doesn't love her anymore and, hopefully, she will move on to healthier relationships that don't have such a risk of hurting her.  He also hopes that by lying to her and making her think that he doesn't want her anymore, she won't attempt to follow him as she would if he left without lying, furthering his intentions to keep her safe from harm that might be inflicted upon her by him and the other Cullens.

After the accident the night before, at Bella's birthday party, Edward decides he must break up with Bella for her own good.  The Cullens had hosted a little birthday gathering for Bella at their home, and Bella sustained a papercut when opening one of her packages.  Jasper, who has more difficulty resisting human blood than the other Cullens, is overtaken and leaps for Bella, who is then shoved into a table by Edward while he leaps between her and Jasper.  She is then cut extensively by glass that shatters when she is thrown into the wall, and must be stitched up by Carlyle.  Edward decides that the risk is too great and tells Bella the next day that he is leaving.