Tools are used to collect Resources from the world around you. Each tool is used for a different type of Resource.
Axes are used to collect Resources from trees. They come in two different types - the Axe, and the Iron Axe. The Iron Axe has higher durability and ATK power, so it is worth unlocking and using when you can.
Pickaxes are used for deposits. Like Axes, there is also an Iron Pickaxe which has better stats. This means you can use it for longer when gathering Resources.
When you are near to a Resource node, your interaction icon will change to the symbol of the tool needed to gather from it, if you have one crafted in your Inventory. There is no need to equip the tool to start gathering - simply tap the interaction icon when in range and your Exile will begin using the correct tool automatically.
Tools do have a set durability, like your weapons and armor. When a tool breaks you will no longer be able to use it, and you will be required to craft a replacement to continue gathering Resources.
You can use tools as a weapon when in a pinch, but they will not work as well as an actual weapon.