When he first started to speak, Katara resolutely decided to let him say what he had to say, then just walk away. After all, she didn't owe him anything, and hearing his life story wasn't something she wanted to do. The more human he seemed in her eyes, the better chance she'd forget how much of a jerk he was, and then she'd end up in the very situation she was trying to avoid.
By the time he'd started to cry - actually, honestly cry - Katara found she just couldn't be that angry with him anymore.
She still didn't really trust him. That was going to take some time. But she was starting to think that maybe he really wasn't trying to pull a fast one on her. Maybe Cordelia was right and Zuko was trying to honestly change. He was just really, really bad at being good. It would almost be amusing, if it didn't affect her so much.
Staring at him for a minute she finally sighed and looked away. "Okay," she agreed slowly, blue gaze flickering back to him while her face was set in a carefully neutral expression. "Let's say I believe you. That still doesn't explain why you left this time."