How can I find free camping locations in the U.S.?

National Forests generally allow "dispersed camping" without any fee. Exact rules may vary from place to place, but typically you may camp anywhere you want on Forest Service land, as long as it is at least 100 feet (or some such distance) from a road, trail, or water source. You can normally park your car anywhere along a Forest Service road as long as you don't obstruct it. Check the web site of the specific National Forest you want for exact details and any possible restrictions.

Of course, this leaves it to you to find a location that's actually suitable for camping, which may be difficult or even impossible in any given area. You'll also most likely have to bring your own water (or be reasonably near a stream or lake and have something for purification), bury or pack out your waste, and haul everything back and forth from your car. If your main goal is a quick rest before getting back on the road, you might find that all this extra time and effort is not worth the cost savings compared to a developed campground. (But it could also be a lot of fun!)

Backcountry campsites are also usually free, and may have established tent sites, lean-tos, outhouses, etc. However you will normally have to hike in a significant distance, so this is not as good for quick overnights.