The bag of materials is automatically added to the tool belt during the Invention Tutorial. It is used to hold materials obtained while using the Invention skill. It is never actually placed in the player's backpack and only appears in a dialogue box.
Clicking it on the tool belt opens the materials interface, showing the types and amount of materials possessed. The same interface can be opened by clicking the "Invention pouch" in the Inventory tab, next to the currency pouch.