What does it mean when your horses number under a level is highlighted Red?

You can find the answers in the FAQs:

https://horseeden.com/faq.php#29

What Do The Red Numbers In My Barn Mean?
The red numbers mean that your horse has leveled up before their rider, and so they are at a disadvantage in showing. It would be beneficial to switch your horse to a higher level rider. However, if your horse/rider is entered into a Federation, you may want to wait until the end of the year before switching riders, otherwise you'll need to re-enter them in the Federation. If the number is green, it means the rider is a higher level than the horse, and so they are at a showing advantage.