Do horses level up on their own?

Or do we need to hit level up? Like for showing?

You need to hit the "use boost points" button. You'll know which horses need to be levelled by the red arrow next to their name when you look at your division. The same goes for your player level. If you notice that the bar is full, you need to go to your player page and level up manually. It'll be the little box above where all your divisions are listed. I don't think you stop earning XP if you don't level yourself up, but there are things you can only do at certain levels, so it's better to keep an eye on it.

Horses increase their grade (Novice, Local, etc) for showing on their own, but you do have to hit a button to apply the stat boost that comes from training.

Thanks guys!