cairne was fighting for what he thought was right. garrosh could give him 100 ways to give up, and he wouldn't take it, because he believed garrosh shouldn't lead the horde.
in the end, he was right. his belief was fueled by wrong leads, but garrosh should never had been left as the warchief. had cairne won the duel, the terrible events in theramore, gilneas and pandaria would have never happened.