Ah, young grasshopper. You are now discovering the ways of the world. It is not enough simply to be right; one must be cool about it, too. Which unfortunately means not getting pissed off that the other person is talking bullshit, because that only makes whoever you’re talking to double down, no matter how pants-on-head their point is.
As to what is wrong with them and why: pride, and embarrassment.
It’s a cliche and also very hard advice to follow, but it’s true that the best thing to do is smile and walk away. Know that they know you know they’re wrong and an idiot, and take comfort in it.
Edit: It’s very hard. I struggle with it every day, including yesterday, and today. Just gotta keep trying, we’ll get it eventually, right?