You can transfer your horse from the Star Stable Horses app by launching the app and clicking on the Star Stable logo on the bottom left corner of the screen. You will then be asked to type in your account details to complete the purchase. You can only transfer your horse when it has reached level 10 and it will have that level in Star Stable Online as well after transferring. Nevertheless, it will cost the same price as a level 1 horse in Star Stable Online!
Your horse will automatically be a level 10 horse when you transfer it to SSO!