How do you handle it when you've botched something? First, you need to know what happens in your manager's head when you make a mistake. Beyond thinking about the repercussions of the mistake itself, she's worrying about what it means for the larger picture: Did the mistake happen because of sloppy work habits or was this one isolated incident? Is there a fundamental problem with your systems or approach to the work? Do you "get" that this is a big deal, or are you shrugging it off and thus likely to let something similar happen in the future?
Once you understand this, the formula for handling a mistake well becomes more intuitive.
And if you want to read that formula, check out the full post, which is posted at U.S. News and World Report.