First, you need the given villager's amiibo card (which, good luck getting a specific one without paying an arm and a leg at this point considering the demand). Then you invite them by scanning the card through the ATM, and they'll appear at the campsite. Then, over the course of 3 days, you need to talk to them and craft items for them to slowly convince them to move in. If your town is at full capacity, you can manually choose who gets replaced by them (this is distinct from random campsite visitors, who choose a villager at random to take the place of).