You can unlock the ability to build the campsite by upgrading your Resident Services building from the original Nook tent to an actual building. It can be achieved by paying Tom Nook the debt you owe from getting to the island, fulfilling tasks set by Tom Nook's two kids (Timmy and Tommy), building a bridge, and getting three islanders to move to your island by crafting items.  Once you get to the Resident Services building (note: it will be closed for a day while being built), talk with Tom Nook. He will talk a LOT about his favorite musician, K.K. Slider, but also give you a recipe for a campsite!  Gather the materials required for the recipe: 15 wood, 15 softwood, 15 hardwood, and 15 iron nuggets. If you don't have a workbench yet, use the one right near where you talk to Tom Nook at his counter.  Craft it, and then choose where you'd like your campsite placed on your island on your island. Then talk with Tom Nook again, and he'll make you wait the rest of the day for the campsite to be built. It's so exciting!!