To chat up is to talk playfully or casually with someone, especially with the goal of convincing them to do something. Your brother might chat up a movie theater worker in the hopes of getting a free ticket.
When you chat someone up, your light banter is persuasive — although in many cases, your goal is simply to flirt, or simply to start a conversation. You could chat up your new classmate, or chat up your seatmate on a long airplane flight. Another way to use this phrase is chat someone up, as in, "I'm going to chat George up and see if he'll give me some of his pizza."