Deports will store military supplies and then automatically distribute them to any nearby armies, if within the army’s command range. When supplies are consumed, they are paid for, and these costs are covered in army upkeep.
New depots can lie built by armies and then upgraded to allow more supplies to be stored. Depots can also be captured and raided by the enemy.
Like industries and armies, the depots connect to the economy via IIPs and will acquire the supplies from the IIPs. If nearby IIPs have low transport capacity, the depots will struggle to keep in stock if they supply large armies.
If you check any of your depots and find them empty, this means that nearby armies have such high consumption of supplies that none are left to be stored in the depot, and the depot cannot refill the stocks fast enough.
You can see further details about connected depots and their capacity in the tooltip provided in the connected depot’s info in the unit panel: