There are a number of benefits to a horse going barefoot. The most obvious is that there is less damage to the hoof capsule from the use of nails. Nail holes will weaken the hoof wall and create an environment for anaerobic bacteria and fungus to get a foothold in the hoof wall and possibly cause problems later on, such as white line disease.
The greatest advantage to the horse is that he will have better proprioception, that is, he will have a better sense of where his foot is, and what is underneath it. A horse in shoes is like you going around wearing heavy gloves all the time. You can still get a general feel for something, but there is a lack of sensitivity. With better proprioception, the horse will be more surefooted and have a better sense of his environment and can make compensations and adjustments for uneven or slippery footing.
In terms of long term health, if a horse is left barefoot, he will develop a better internal structure within the hoof and is likely to stay sound longer. The effect of a shoe on a foot is similar to putting a cast on your arm. Since the structure cannot move as it was intended, it will tend to atrophy, much as the muscles in your arm will atrophy due to a cast. The most damage is done when a horse is shod at an early age, as are many racehorses. This will not allow the internal structures of the foot to develop while the animal is growing, and in some cases will cause the animal to permanently have poor quality feet.
A further advantage is for the social health of the horse. If a horse is shod, particularly on the hind feet, it is dangerous to turn them out with other horses, due to the possibility of an injury due to a kick. When barefoot, horses can be turned out in groups which is much better for their social health, as horses are herd animals and generally do better when they are allowed to interact with other horses.