The Stable is a way for you to put duplicate Familiars to work! By boarding extra Familiars, you can increase the stats for all Familiars of that type you currently own.
Tap Stable from the Familiars menu.
The Stable will list all currently boarded Familiars. Select Board to add more Familiars. The type benefits max out at 5.
Keep in mind, when a Familiar is boarded within the Stable you cannot use that Familiar in battle. However, Familiars can always be released from the Stable to use again. Doing so will cause you to lose the bonus you gained for having them in the Stable until they are boarded once more.