Damon asked Emily to protect Katherine at any cost, after the founding families took Katherine, right? Then Stefan and Damon tried to save Katherine, but died in the process. When they woke up, Damon thought Katherine was dead, he thought that she was burned in the 불, 화재 (which means he thought that Emily spell didn't work), which explains why he wanted to die. But here's the tricky part. I recall Damon telling Stefan, "You know Emily waited till after I turned to tell me that she was successful protecting Katherine with her spell, she didn't want me to know about the tomb...."