My clients, and just casual observers of my work, ask me this all the time. How did you get that shot?
It's patience, mostly. And being super comfortable around horses.
Think about the last time you had YOUR photo taken. A family photo, wedding, professional head shot...The photographer likely guided you to stand in the best light, tilt your head, look this way, and other instructions before they took their shot. You can't do that with horses. They don't stand in the right light, so I have to wait for them to move and I have to be in the right spot when they finally get there. They have their heads down a lot, sniffing at what was there before them and looking for food, so getting them to look at me can take some time, too.
Many times my clients will call to their horse or wave their hands. It doesn't generally work, and they quickly get tired. I find that if I introduce myself and my camera, if the horse is really interested in interacting, we'll get rolling pretty quickly. It's easy to tell if they get what I'm doing and are going to partner with me or not.
Very rarely do I get a subject that is truly uninterested. Most of the time it's the exact opposite - they're in my pocket and so close that my camera can't focus on them. This is where being comfortable around horses comes in. They have no concept of their size in relationship to you, nor do they realize how big and bony their faces are. Knowing where to stand, when to be firm and when to be gentle, makes a difference Most of the time, I find myself sitting in the dirt in the middle of a round pen, arena or their stall, just letting them be a horse until the shot comes.
Taking photos of horses takes longer than taking photos of humans, but I don't mind. There's no place else I'd rather be.