Refer to "Crow’s Nest" on how to find pirate hideouts. To destroy a hideout you will need a strong convoy (8+ warships with maximum sailors and weapons). Highlight the hideout on the map and press to engage. You will have to fight 2 pirate convoys first which shouldn’t prove difficult with a strong convoy. After that you will have to land and fight the pirates on land. With a full crew and weapons this will be easy. Once you have destroyed 10, the achievement will unlock.
TIP: You will have to destroy a few hideouts in the Adventurer campaign and these will count towards the total.