Another reason for airing down is that your tires will deform and get better traction on large rocks and especially on granite slick rock out west if they have lower air pressure. I have seen some actually wrap themselves around the curve of a rock. Too low of air pressure can however cause them to come off the wheel rim. I have gone down to as low as 15 psi when I used to do more hardcore rock crawling with my last TJ on trails like the Rubicon, Dusey Ershem, or Kaiser pass in California. As Jon mentioned the best ride comes from lower air pressure where the tire is not rebounding from the rocks or trail.
Many of us carry air compressors mostly for airing up. We actually had a give away of free air compressors on the first Sedona Trail run.