It was his turn to tell a story about his own past. Hermione was surprised that he was willing to be so open and honest with her about his vulnerabilities. These were facts that could be used against him if in the wrong hands. By Kingsley telling her something so personal, he was offering his own proof in a way that he could be trusted. Most of her concerns and worries about him weren't likely to go away just because of one story time, but she appreciated the gesture.