Before you can enter your horse in a race, you will have to move them to a "home track". This is the track they will be based out of for training, so it will become a track they are familiar with. When you first joined, you selected a home track, which is initially where all of your racing string will have to be located. However, not all of your horses have to have the same track, and they don't even have to have a home track within your stable's region. You can go to the Racing tab and hire an assistant trainer at the Office, which allows you to choose another track to base your stable out of. For example, if you started with Coldstream Downs, you could add Werburgh Racecourse and have a racing string based in the east AND a racing string based in the west to help split up your horses.
Home tracks will do a couple of things for horses when it comes to races. One, they will determine which region the horse lives in. When a horse races outside of their region, they will lose more energy than normal due to the stress of traveling. This will not negatively impact them in that race, but could make them too tired to race in the following week. Secondly, horses earn a bonus to their race score when they are racing at their home track. Think of it as a home track advantage, so to speak. Horses take time to adjust to a new track when they move, thus, moving them just before a major race is NOT going to benefit them. They can also only move home tracks once every 6 in-game months.
To set a home track, go to a horse's page and go to the Manage page. There you will find a section that lists their current home track and has a dropdown to select a new one. If it says Locked, that means the horse has moved too recently to be moved again. Once you've successfully selected a home track, it will also appear on their main page in their general info section.