Jaime, Brienne and Cleos are get out of their boat at an inn expecting it to be empty, but find a man who is not an innkeeper who serves them horse steak. He also sells them horses so they can continue on land which might be safer. At this point Westeros is basically like Somalia, with warlords fighting it out over territory and bands of rogues roaming in between and anyone travelling amywhere is going to get killed.
They take a different route to the one recommended by the man who is not an innkeeper in case he is sending them into a trap.
We get to spend some time in Jaime's head. Basically the whole plot up to this point is driven by his and Cersei's love for each other. Tywin was going to marry him off to a Tyrell daughter so to stay close to Cersei he became a kingsguard. This didn't work because it annoyed Tywin who resigned his post as hand and left with Cersei to Casterley Rock. (The law of drama suggests that Tywin is going to find out about Jaime and Cersei at the worst possible moment, at some point later in the books.)
This left Tywin free to plot against Aerys, and it seems like it's partly Jaime's frustration at the separation that leads to him killing both the new hand and the king Aerys. If Jaime had just married Tyrell's daughter, Tywin would be hand, Cersei would have married a Targaryen (maybe Vyseris), and none of this would have happened. Probably.
