It can be used to organize the Vault Dweller's inventory. A backpack can contain any items that total less than about 40 lbs, but cannot contain larger items like armor.
To use the backpack, right-click to get the command cursor, then left-click and hold on the backpack. Move down to the open hand icon and release to open the backpack. The player will know the backpack is open because its icon replaces their own. Drag items from either hand into the backpack, then click the icon (where your character usually is) to return to the normal inventory.
Alternately, simply drag the item over the back pack; or if in the backpack, drag the item to the icon (where the character usually is) to move it back to the main inventory.