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Revealing mistake: At the beginning of the episode, Robb and Blackfish Tully are pushing Hoster Tully's funeral boat down the river. When they climb back onto the dock, there is already a trail of water going from the steps and ending where Robb ends up standing. None of the other characters standing on this portion of the dock have wet clothes, meaning none of them went into the river, indicating that the trail of water was from previous takes.
Answer: Basically it's because he broke his vow. A member of the Kingsguard is sworn to protect the king at all costs. Jaime elaborates more on the deed to both Catelyn Stark and Brienne of Tarth, telling Catelyn that no matter what course of action he took, he would be breaking one vow or another (i.e. if he obeyed the king, he would conversely be disobeying his own father), and telling Brienne that the Mad King was planning on burning all of King's Landing, but he did not bother to tell Ned Stark that. Ned Stark felt that killing Aerys was dishonorable and excessive. Robert Baratheon still could have usurped the throne without Aerys being killed.
Phaneron ★