Your player level increases by doing quests you can find in the hoofbeat plains (exploring).   At this time leveling up as the player does not have any benefits or rewards, just bragging rights / Hall of Fame listings

Is player level or horse level needed for breeding rare horse breeds?

Unless you want to breed Chincoteagues, Marwaris and Akhal Tekes? These three are considered rare breeds and can ONLY be bred once the horse level reaches 25. Even if they hit age 3, if you go to the Heart icon on their page, it will state it cannot be bred until level 25.

Player levels do not parlay into this at all, nor do horse levels outside of the named breeds above. As long as they are not a Marwari, Teke and Chincoteague, there is no level requirements.

Oooh how interesting! If we do a crossbreed campaign, can we choose to make our new breed a rare breed? I'd want to have a minimum of level 50 for breeding, and limited to 5 foals per horse or something like that