"It is a good question why I caused this problem. It is weird to think there is a competent company that has been around for so many years, yet they have no procedures in place to stop this from happening. You would think that any changes that could cause downtime on a clients website would go through an automatic test suite and only after passing all tests would it be tested by human QA and finally made live. In this particular case I am the one who has made a mistake, and being a sensible person who is eager to constantly better myself, I will try and learn from this error, however I work alongside dozens of other people on my team who like me, have every chance of making a mistake at some point. It seems like a bad policy for us as a company to say that we should expect every person on the team to break a clients site and then rely on overtime from other people to fix it. So of course I will try and do better, but this is not fixing the root of the problem, the root of the problem is something that needs to be fixed at a much higher level. Have you, as my boss, not considered this problem already? What has the company learned from this? I would be happy to be part of the team that solves this problem by creating unit tests and creating policy to avoid this."
Of course I purposely make this slightly stand-off-ish, over the top, and written from a very specific perspective, to illustrate a point. But the reason I do this is I absolutely agree a single developer shouldn't feel bad about making a mistake (they should try not to, but these things happen), but the company should put measures in place to minimise the impact of mistakes. When the company fails to do this, the company has screwed up a lot more than the developer.