When something percolates, it spreads gradually. If the news of your victory in the spelling bee spread slowly through your entire school, you could sit back and enjoy watching it percolate.
When a liquid percolates, it's filtered through something, and when an idea percolates, it's filtered through a group of people. Percolate often refers specifically to coffee, which is brewed by mixing ground coffee beans with hot water and filtering them through paper — in other words, coffee percolates. The Latin root is percolatus, which comes from per, or "through," plus colare, "to strain."