Many of us will learn how to ride and how to do fancy movements or jump high. Few of us though will learn how to train a horse. I mean really train them to do something from scratch. We take lessons to learn new things but for the most part the horse knows how to do it.
I can speak from experience that there is nothing more rewarding than seeing a horse you trained succeed. Learning the process instead of just the aids gives you an edge in your riding both at home and at a show. Training a horse gives you more ways to get the same result because you try different methods until it clicks with the horse. Think about the training process like a maze. It can be confusing, frustrating, and there are many ways to get the the other side. One you are out though, you feel accomplished and self-confident.
Being able to take things step by step and hone a horses ability will make you a better rider both physically and mentally. Slowing things down and figuring out what needs to be done is an amazing skill to have. It also shows you what you are made of. Do you give up when you hit a wall or do you find a pick axe and start working through it?
Training a horse also helps you develop your riding technique. Riding a green horse always keeps you on your toes. When you first start off, you could have a different horse every day. Being able to hope on a ride the horse you have is priceless! You never know when your level headed horse might decide they want to be a dragon at a show. Your riding styles needs to be adaptive and flexible while remaining consistent and reliable.