She asks if he knows Illyrio, the guy who gave her those eggs. He does. She points out that they hatched. "When I stepped into the fire, my own people thought I was mad. But when the fire burned out, I was unhurt. The Mother of Dragons. Do you understand? I'm no ordinary woman. My dreams come true." She's leaning pretty hard on the "mother of dragons" angle. Look, just because you have some exotic pets does not mean you're going to conquer an entire kingdom. Spice Kingpin says, "I admire your passion. But in business, I trust in logic, not passion." Then he calls her "little princess," which she finds demeaning, because she'll take what is hers with fire and blood. "Yes, my lady. But not with my ships."
Arya cleans up Tywin's table during one of the rare moments that it's not surrounded by people being yelled at for being incompetent. She sees a note that says "Robb Stark" on it, but we don't have time to read the whole thing. Tywin appears behind her and asks who taught her to read. She says it was her father. Tywin reminisces about the time he tried to teach Jaime to read, but the letters kept reversing in his head. So that's why it's spelled "Jaime" and not "Jamie" like I want it to be! Back then, a Maester told Tywin it was a waste of time, but Tywin sat Jaime down for four hours a day until he learned.
Tywin asks after Arya's father, and she says he was a stonemason who taught himself to read. And then he died. Tywin continues to exhibit interest: "Quite a man. What killed him?" "Loyalty." Tywin asks, "You're a sharp little thing, aren't you?" Arya starts to ask a question and then says she shouldn't. Tywin agrees, but lets her finish. She asks if he knew his father. He did! He grew up with him and watched him grow old. As he reminisces about how his father was a weak man who nearly destroyed the Lannisters, he sits down facing away from her -- the better to look wistful. Once he's turned away, she pockets the note. Then he says he's cold, so she's sent off to fetch wood.
In a nook in the castle, she reads the note. We can't see all of it (and you have no idea how much that bothers me), but here's what we can make out: "Marching ten thousand men on the tooth. Estimate to reach you by week's end. (SOMETHING) Robb Stark moving troops south by coast. Alert (SOMEONE) turn east at Silverhill." So it's something important! She runs downstairs, presumably to send an important message to Robb and runs into someone who wants to know where she's going. I don't actually recognize him, but I happen to know that it's Amory Lorch. She tells him she's on her way to the armory on Lord Tywin's orders. He's not particularly convinced. And he can see that she's trying to hide the note. She says that Tywin gave it to her. He doesn't buy it. She runs. There's a chase! She escapes, hiding behind a pillar.