Humans have an inherent need to connect with nature (biophalia). The horse, as a large prey animal with a long history of being a domestic animal to humans is a natural course of people to connect. The theories behind why equine assisted coaching works is because people drop the barriers they carry and have to focus on being in the moment. Working with such a large animal that is always sensitive to it’s surroundings and other natural objects pushes humans into that same state of mind. 
Clarity emerges when the daily busy mind slows down. The horse is a great partner in helping make that happen. Horses bring us into mindfulness of being present minded and in the moment of what is occurring right now. There is no excuses, blaming or stories when you are with the horse, you just are there in their presence.
Animals in general have long been a source of comfort for humans. Simply look at the bonds between humans and dogs/cats. Most dog/cat owners would tell you that the thing the animal brings them calm and even happiness. The equine assisted coaching experience manifests a similar result for most people.