I built a camp site as one of my community works projects but I can't go in it and no one else is ever there. I finished paying for it and then a day later it said it was done but it doesn't seem done.
There are only a number of public work projects that you, the player, are able to use. The campsite is not one of them, as it has been said. You may be the mayor, and you live in town. I'm sure you don't need two homes, right? It looks incomplete simply because nobody is using it- there's a lamp, some boards and what looks to be fabric for setting up a tent. I assume the animals visiting set up and put down after. ;o
Occasionally, random, villager type characters will visit for a day. You can interact with them, and it's even possible obtain special items from them. They play games with you, and it's a good way to fill your catalog or sell some crappy items for high prices. (One guy bought my almost broken axe for 4k bells!)
You can only convince the camper to move in if your town has room, or, one of your villagers is planning to move out. They will move in soon after the other said villager is gone. On rare occasions, you can convince the camper to move in without either of these prerequisites, and this talk will actually trigger villagers in your town to try and move out. They will 'ping' you, come up and ask to move. You can decline, and possibly the next day or within the day another villager may request the same, and you can continue to decline until you get the villager you want to move to leave town.
And as I said, the campsite does have other uses, and according to multiple sources, there are some items you can only obtain from campers, and so it's beneficial to have a chat with them!
The campsite in New Leaf is actually not something to use as-is. Occasionally, characters who don't live in your town will camp there, putting a tent where the campsite normally is. If you talk to that character enough, you might be able to get them to move into your town. Past this, the campsite has no use and can easily be considered ugly or to look unfinished.
If you feel like reading into it more, here's a wiki page.