Persuasive authority means that someone is trying to convince you to do something or believe something based on their opinion or experience. Think about when you go to the playground and your friend tells you that the swings are better than the slide because they had more fun on the swings. They are using their experience to try to persuade you to pick the swings over the slide.
In the grown-up world, lawyers and judges also use persuasive authority. They might use previous cases or legal opinions to try to persuade other judges or lawyers to agree with their argument or decision. Just like your friend used their experience to persuade you, lawyers and judges use past cases and legal opinions to persuade others.
Sometimes, persuasive authority can be really strong and change someone's mind or decision. Other times, it might not work as well, and people might stick to their own opinions or decisions. Just like on the playground, everyone has their own preferences and opinions, but persuasive authority can help influence those decisions.