Each time your horse advances to a new level, it will receive some skill points that can be used to increase some of the skills.
The skill points will increase the chances your horse has to win a show (or to place in the top 20). The skills are directly related to show types. For instance a higher speed will definitely be a bonus if you enter a Racing type, but it would not matter in a Dressage.
Skills go hand in hand with specialization. If you plan to specialize your horse on a few show types, then you should only raise the skills that are related to those show types.
(More on that will be added later)