If you recruit an advisor from the cultural tradition screen it goes into the pool of available advisors. One year later, assuming you didn't hire them, they become available for other nations to hire, triggering the message you are asking about. There as also a possibility of an advisor randomly spawning in one of your provinces that will behave in a similar fashion. I don't believe there is anything you can do to increase the chances of this beyond having a good reputation (low infamy), which seems to help, and a large number of provinces.