There is usually an official checkout time, like a hotel. After that time, you can just show up and take it. If the previous campers leave early, you should be able to claim it as soon as they leave. You normally fill out an envelope, put your money in it, put in some kind of box, and attach a “receipt” near your campsite to show that the campsite is occupied and that you paid. Campgrounds usually limit you to staying 7 or 14 consecutive days.