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Human naturally have pride and ego and do not want to be told they are wrong and have made a mistake. Except for human beings who are objective, logical and open to new ideas, some feel embarrassed when you correct them and they feel like you are trying to pass a message across which says you are dumb, I am smart. Some feel that you correcting him is a negative attitude.

Lack of maturity is also one of the reasons people get angry when corrected. Some want to win am argument at all cost even when it is glaring that they are wrong. They believe you are the one missing the point they are trying to make.

It however also depends on the manner with you the person who is correcting is making the correction. If you rub off as very knowledgeable it might be offensive. Correcting someone in public can also be embarrassing leading to anger. Also when the person correcting is known to always want to correct people, this can also get the person being corrected pissed off or angry. The tone of voice used to correct someone can also be a reason why people get angry when corrected.

People get angry because we naturally learn by rebellion. This is the reason why, when you tell someone not to do a thing, everything in them will push them to do that same thing (Eve and the Apple) (a kid and naked flame i.e candle light).

It takes a trained mind and a humble personality to always accept when you are wrong or even look away when others are wrong to maintain peace. Plus some mistakes are as a result of rooted believe. So, you would have to address the believe before the mistake.

More out of embarrassment. Sometimes it is best to ignore them and guide them carefully not always highlighting them.