As a professional agility trainer, I can say there are a ton of benefits to agility. In fact, the character limit won't let me begin to list them.
First for me personally is the strong bond I develop between my agility dogs and I. The higher levels we train to, the more that bond grows. There is nothing like the bond that has developed from the literally thousands of hours of training and trialing it takes for a dog to reach an agility championship in the harder venues like the AKC or USDAA. The handler and dog become so in tune with each other that they can each read the subtle non-verbal cues the other is giving way beyond the ability of any house dog/owner. My vet is very used to me showing up at his doorstep, plopping my dog on his exam table and saying, "I don't know what's up, but something's wrong." After an intense examination, he'll find something amiss - something small usually. He often says that his other clients would never have seen that, and that he wouldn't have caught it in a normal exam. Conversely, my dogs are capable of picking up on my slightest non verbal cue. The bond between my top level agility dogs far exceeds any bond I experienced with my other dogs, even my competition trained obedience dog.
The health benefit that have been mentioned here exists, but honestly, it's a side benefit compared to the other benefits. A house dog can get in shape without agility, and actually agility isn't enough exercise to get an agility dog in shape.. Agility dogs need to be "cross-trained" to be in top shape for their sport. Just practice, training and trials alone aren't enough. So while health benefits exist, they aren't as strong as someone who doesn't do the sport would think. In fact, agility dogs are at higher risk of injury, perhaps negating some of the health benefits from exercise.
The best benefits to agility need to be experienced to understand. They cannot be just "listed" as they tend to be intangibles. For instance, one of the great benefits of agility are the friendships made. I have hundreds and hundreds of great agility friends. We differ in all other areas - politics, religion, background - but we have dogs and agility in common. This binds us and brings two people together who would normally have never found common ground. The health benefits to me both mentally and physically really cannot be weighed both from the physical exercise of handling and the social side. No room left - much more to list.......