A military stable would also have a training grounds, barracks for the troopers, blacksmiths, armories, and the like. A horse must be highly trained to be successfully used in battle. Any civilization that wants to employ a large number of horsemen are highly encouraged to invest and build as many stables as possible.
Horses are raised in the stable, and horsemen develop along with the stable's level. The Stable allows Horseback Riding to be researched and is also requred to build Relief Stations.
The in-game description of Stables claims "the higher its level the faster troops are trained." Perhaps it means that faster horses can be trained with higher level of the stable. It does not increase speed of training however. Building levels do not affect speed of research/training/construction in the game.