Put it in relative terms. Basketball = Life for these guys. It's their jobs, they come out and play for months hoping they might win the championship. Only player in the NBA who might have more pressure to win a championship on them besides LeBron is Kobe. After all this hard work, and everything going great, he loses in the Conference Finals, one step away from a championship.
Imagine you've been working for months on end trying to get a promotion. You've got the best resume for the promotion (Cavs easily had the best resume for the ship) and then someone who had a great resume but not quite the same as yours, comes in and gets the promotion over you. How would you feel?
Not saying it was right that he walked out without saying anything. But he certainly didn't "storm" out of the Arena as many people said, he just simply didn't talk to anybody after the game. He actually congratulated Dwight before the game was even over, but people don't even know that.
I disagree with him when he said he isn't a poor sport though, because what he did is by definition a poor sport. I think he made a mistake, and wouldn't label him a poor sport because of this one incident though. I can only imagine how he was feeling at the time.