The horses' attributes (speed, jumping power, endurance and agility) are mostly important for fulfilling guest's requirements or for the events you can send your horses to later in the game.
For example, a guest might want to hang out with a horse that has especially high speed, or a jumping event may require that your horse has at least 8 jumping points to compete. They don't have a bearing on the day to day activities around the farm though. You can improve these values by sending your horses to train at the jumping course or the indoor riding arena.
Horses may also have some special characteristics denoted by the white and black symbols below the regular attributes - some horses are especially calm (zen symbol), while others are very friendly (sunshine). Just like the main requirements, these are simply for fulfilling certain event and guest requirements. They can be passed on to offspring, and certain combinations can even yield new traits that you won't see on horses in the shop!