In many ways, horses are a vehicle like any other technique in Battlefield 1... They have their own unique traits, but the bottom line is that horse riding isn't too different from driving a tank. When you mount a horse, you will lose access to your usual equipment, instead of which you will receive a rifle and saber. The rifle is good at long range, while the cavalry saber is very useful in close combat.
If you want to get somewhere quickly, send the horse into a gallop, but remember that you will not be able to accurately aim. Since horses are quite large and have no armor (unlike other vehicles), you must use a cover if you want life for your horse. However, she has a lot of health, so a few bullets won't do much harm to her.
Also, your horse can jump, but the trick is to figure out which obstacles he can jump over, and which ones will be too high. Riding a horse allows you to throw ammo and first aid kits to your allies, so you can become a mobile support point. You will have fewer options than a Medic or the same support class, but you will move significantly faster.