The equestrian centre's manual
3.4 - The rosettes
3.4.2 - Equestrian centre rosettes
Rosettes are rewards for horses that finish first in competitions that have more than 75% prestige. Winning a rosette gives you double the Equus!
Rosettes are a good way to make your equestrian centre stand out, because they're shown on the special rosette page for all horses that have won them.
To activate your rosette, just go to your competitions page.