Jaime Lannister in a nutshell.
His face in the last gif says everything. He doesn’t understand how stopping a mad murderer isn’t justice. He doesn’t understand how someone, after knowing what the Mad King did, can still hold this notion of a king being sacred, irresponsible and untouchable. He thinks he did the right thing, and he knows no-one else does, but he doesn’t understand that nobody understands.
He’s trying to tell him that, to make him understand, to say he’s sorry for his father and brother. Even if it is in his smug, annoying way. He knows Ned Stark doesn’t like him, and doesn’t care, but ultimately he can’t cope with Ned being mad at him for killing a king.