The Mine is where you can use the rewards from Cargo Ships and get ingots in exchange of them. When you get Pickaxe, Dynamite, Bomb and TNT as reward from Cargo Ships, you can use these at the Mine to get ingots.
In order to get ingots, tap on the Mine, drag a Pickaxe, Dynamite, Bomb or TNT on the Mine. It'll blow up and give you ingots in exchange of it. You can use the ingots to upgrade your buildings. You can upgrade buildings only with ingots and ingots can only be gained from the Mine.