What does Horse equipment do?

All of the horse armor in the base game is cosmetic. However, the Champions' Ballad DLC Pack 2 added two pieces of equipment which give your horse new abilities.

• Ancient Bridle: Gives your horse two extra spurs of stamina.
• Ancient Saddle: Any horse equipped with this saddle will be summoned when Link whistles, provided they are not boarded at a stable and are in an area where link can ride them.

You can earn these items by completing the EX Ancient Horse Rumors side quest. Aside from these two items, any other saddles or reins are purely cosmetic.

As Yuuki mentioned, you can change your horse's saddle, reins, or mane style by visiting a stable and talking to the person tending to the horses.

The saddle and reins are for appearances only. If you don't like how it looks, ride your horse to a stable and find the person tending to the horses. Upon talking to them while riding your horse, you will be asked whether you want to change your horse's appearance.

You will be able to change the saddle, reins, as well as your horse's mane style.