It is at the stable where you keep your horses for your army to use. You are allowed to produce horsemen when you complete building a stable. There are no bonuses given by the stable to increase production time of cavalries, but you need it as a prerequisite for your relief station.
High level stables also allow you to research and level up the horseback riding skill. Since we all know that the horseback riding skill allows your cavalry to have better riding speeds. Usually you will only have a stable in one of your cities after you got your relief stations leveled up to max.
You need a Stable as the research requirement for Riding technology. Riding technology will increase the moving speed of some of your troops faster.
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