Why do people often find it difficult to admit that they are wrong?

We sometimes have trouble seeing we are in the wrong. 
Sometimes it’s just a matter of misunderstanding. And sometimes the psyche defends itself, shifts priorities and subconsciously makes the mistake seem less important – “it wasn’t that big of a deal, so it’s not important to acknowledge”.
When we do realize we are in the wrong, Ego goes to hyper-drive and has to deal with a form of failure.
 I must admit, first to myself, that I am wrong.
 Then I have to admit it outwardly.
If the argument was drawn, if it involved hardship and emotional turmoil, it gets even harder. I may have to apologize for what I’ve said or done, for any outburst, for not seeing the other point of view to begin with.
Each step requires more and more to deal with, it’s hard on people with any insecurities (like, everyone, at some point or another).
On one hand, you should always be prepared to admit you’re wrong.
 On the other, you should still have some confidence in your opinions and knowledge.
 It’s hard to find the balance between the two. I’d advise to the other side of the argument, to keep in mind the trouble the other side is having in admitting. Don’t slap them with facts, they’ll just get even more defensive.
Be tolerant of their difficulties to understand/accept, and help them through it. 
And always remember that you could be wrong, too.