Start by Admitting your Mistakes
When someone is angry at you, organize wisely what you wish to say. Begin speaking in a manner that is likely to have a calming effect. For example, begin by admitting your own mistakes. When you start off in an appeasing manner, the person will pay more attention to your words, and this will prevent him from causing you harm or loss.
We find an example when Abigail successfully calmed down King David, who was furious at her husband (see Shmuel 1, 25:25). She began by admitting that she herself had made an error. Only then did she present her arguments to King David. When you concede that you are wrong, others calm down.
When someone is angry at you, and you start out by either blaming him or denying it, you will usually increase the person's anger. If you want someone's anger to escalate, the best way to do this is to either say: "It's your fault, not mine."
It takes courage to admit your own mistakes. Even if you are only responsible in a small way, it is still best to start off by saying something like, "Yes, I could (or should) have done differently. I'm sorry for any pain or inconvenience I have caused you."
This will put the other person in a calmer state, and he will then be much more likely to listen to what you have to say in your own defense.