Why Air Down Your Tires?

A brief lesson.

http://www.toyota-4runner.org/off-roading/102636-airing-down-help.html

Under most circumstances, it's important to keep your rig's tires properly inflated at all times. On pavement, appropriately-inflated rubber ensures a flat contact patch ultimately providing optimal traction - which is what you need to stay safe.

However, when venturing off-road where the ground is no longer even, the need for a flat contact patch goes out the window. The thought process is fairly simple: The less air pressure in the tires, the more they can conform to the terrain. Your tires can wrap around rocks and other obstacles because having less air pressure makes the tires more malleable. 

http://www.clubxterra.org/forums

A Simple Illustration

The following three videos from do a great job at explaining this further:

As mentioned in the videos, airing down your tires too low can cause the tire to come off the wheel which can make for a bad day, especially for a novice.

tr4.org

Beadlock wheels are the go-to remedy for this problem but they're not for everyone. Beadlocks aren't even street legal in all states and are typically reserved for the most extreme trail riders. With that in mind, if running traditional wheels and off-road tires, you want to be conservative in how low you air down your tires. Also, airing down your tires creates the problem of airing them back up when you are finished on the trails. The best (but fairly expensive) solution for this is an onboard air compressor.