>Kyo get so angry?

I didn't really pay attention to either of the hat flashbacks but when I rewatched the series this was the same episode he said he hopes Tohru finds someone she can depend on and open up too, I'm pretty sure Yuki said what he said so that Kyo would pay attention to Tohru and tell her how he really feels by creating a little competition, in the first season she tells her friends she could probably fall in love with the guy who gave her the hat or at least be good friends with him.

>But why is he thinking of this anyway?

Not sure why he probably regrets ever saying it or maybe the hat was given to him by someone special probably his dead mom and he feels bad for letting his hate for someone else hurt him so much.