You can see your horses' ages in the stables if you click on the house button and then the Info "i" on the individual horses. This will open up a window with their stats on the right. The age display is the one with the cake. You can see these stats in other places too (for example when you assign horses to buildings).
There are 5 age stages, each of which lasts a certain amount of time, beginning when you click on the cake symbol for a new age stage:
1 Bar - Foal - 30 minutes (begins at birth)
2 Bars - Juvenile - 96 hours
3 Bars - Adult - 192 hours
4 Bars - Senior - 96 hours
5 Bars - Retired
Once a horse hits the last age stage, you won't be able to use it for anything anymore. You will have the option to either make it young again for Rubies, or send it away.