Why do people have a problem when someone tries to correct them?

I have my cousin who is very arrogant. I understand he is very confident and extremely smart and successful but I guess with that came arrogance like he always wants to be in the center of attention. I tried to correct him one at times and disagreed with some of the stuff he was insulting me and saying negative things to me as if he is showing hate and belittling me. I felt so bad