The Mine can be found on the Island Farm. When you unlock it, you can find valuable raw materials that can be used to make other products, e.g. Copper Ingot and Gold Ingot. Which raw materials you find depends on the selected explosive.

Usage: Tap the mine if the bar has above 1% capacity. To set off the explosives in the mine, drag the explosive material that you want to use to the place holder slots.
Each explosive shows how much % of capacity it needs to be able to be used - check it near the mine miniature icon near the explosive.
Tap the book icon near the explosive to check what materials you can get by using this explosive.
You can craft explosives in the Workshop and Sawmill - tab button "Craft" near the explosive to check.

Also you can invite a neighbor for help by pressing the neighbor button next to the place holder slot. This player will get a Mine Bag as reward for help. Note: every farmer can be asked for help only once in 24 hours.

When all explosives are selected, press on the "Boom" button.

After setting off the explosions you will be able to collect the materials.

At the bottom of the mine you will see a loading bar that indicates the state of the Mine.
If it is zero you will need to wait until it is recharged before you can explode it again. The different explosives have different effects on the mine: the heavier the explosive the longer it takes before the mine is recharged again.