Recreation.gov lists camping and RV sites at U.S. national parks, forests and other federal land. The nightly fee shown is per site, and you're allowed up to the maximum number of people shown for that site. (I've stayed at quite a few of these campgrounds.)
When you actually go to book your campsite, the web site will tell you the maximum number of people allowed to camp on the campsite. Your fee covers that many people. If you want more people than are covered by a single campsite, you'll need to book an additional campsite.
Note that at this particular campground, all of the sites are designed for RVs and allow up to 8 people. If you were planning to do tent camping, you may find that space for your tent is limited, as these sites are listed as having a gravel surface. If you're RV camping, check the details for each specific site before you book, as some will accommodate longer vehicles and some only shorter vehicles.
In other national parks and forests you'll find tent sites, or a mix of tent and RV sites. A sample listing from Yosemite National Park: A tent site allowing up to 2 vehicles and 6 people.
When you check in, the person who booked the campsite must be present and show photo ID.
Also be aware that sites within National Parks will almost always have a separate entrance fee to enter the park, while most National Forests do not have entrance fees. This is usually around $20 per car but it varies by park. This particular one is in a national forest which doesn't have an entrance fee, but it's something to be aware of. If you plan to visit several national parks you may find buying an $80 Annual Pass is cost effective for covering the entrance fees, and the (senior and disabled) passes also sometimes give you other park services for free or at a discount. You can buy one in advance, or get one "at the door" at the first national park you visit.