Final answer:
To get a campsite villager to move into 'Animal Crossing: New Horizons', ensure an open plot is available, repeatedly talk to the visitor at the campsite, and potentially play a game to replace a current villager.
Step-by-step explanation:
To get a campsite villager to move in on Animal Crossing: New Horizons, you should first ensure you have an open plot of land available for them to move into. When a villager visits your campsite, talk to them repeatedly until you can invite them to stay on your island. If your island is full, you may need to convince them to replace a current island resident. This is done through a series of dialogues, and sometimes a card game, which if won, allows you to suggest which villager they can replace. It may take multiple attempts to persuade the campsite visitor and manage which villager they replace. Once agreed, the campsite visitor will move in, and the current villager will move out.