Truth is a funny thing. The absolute truth is that being in a physical body is a lie since we are all everyday Angels. Truth, then, is relative to perspective. The physical realm is something like a conglomeration of the agreed upon truths of its inhabitants (co-creators), so when you are talking about somebody being 'wrong' you may actually be experiencing a few different things.
The most frustrating would be a person who is simply not in tune with the majority and refuses to be. This happens when the majority perception has wounded them somehow. You can invite them back of you like by helping them heal the wound.
A very common instance is when the other person is more in tune with the agreed truths of other groups which you, personally don't run with. You can play red rover if you want that person in your experience.
The most terrifying can be when you are the one out of tune with everyone else. Terrifying, that is, if you haven't learned a few tricks to keep yourself together.
So as for why people 'can't admit they are wrong,' understand that from their perspective, given all that they have experienced, they aren't. Just as from your perspective, with all you have experienced, you're right too.
If you want that person in your experience, find a common ground and build on that. If you don't take it as a blessing that they are so wrong.