I just picked up a mattock on the island, how do I use it?

Tools are an important part of your quest, especially on the island. To use them you first have to open your inventory by clicking the bar at the bottom of the screen containing the scene's name:

And then clicking on the chest:

Now your inventory will open on the right side of the screen (On mobile you can double tab the right bar to make it pop out.)

Simply press the use button to use your tool. After using it the tool will go on cooldown for a while before you can use it again.