What exactly do you mean by horse points? Can you show a screenshot of what you mean? 

if you're talking about experience points, they level up your horse. the levels are mostly meant for breeding, since the amount of levels the mare and stallion have will determine how high the foal's training boost (TB) is after completing foal training. the higher the TB the more performance stats the foal gains from discipline training.

if you mean performance stats (PS) they determine what rank and tier of a discipline your horse competes in, and are raised by training in a discipline or career.