The base is where you can assemble your armies before you go out on battles. The higher your Base is, the more troops you will be able to send at one time. You can also go to the alliance tab in the base to make your city and allied city to your league mates. By making your city allied, they can send troops to your city if you are under an attack. Both league mates have to have their cities allied cities in order to do this.
When you enter your base, you have the ability to send a march out, or see whom is in your allied garrison. The troops you see below are waiting to be commanded by one of your hero's. Click on the plus signs and assign your hero's command over that troop. Do it to all four troop types.
As seen below, you must put a checkmark to allow Allied Garrisons. Once you have done this, your league can send you troops to help defend if you are being attacked, or they/you can store troops there.
You can also recall armies here that you have garrisoned at other cities.