Left-clicking an entity with a badge will bring up a GUI where you can input any items in 9 different slots. When the mob is killed, it will, in addition to normal drops, also drop everything that has been applied to it with a badge.
If a mob is wearing a backpack, the badge will give you 18 slots to put items into, instead of the normal 9.
This is compatible with many mobs from other mods, but it won't work so nicely with every entity. For instance, item frames can have extra drops applied to them, but they won't actually drop those things, making the items essentially thrown away.
If a player gets a badge on survival mode somehow, one can easily increase home storage capacity by adding items as drops to their farm, or the paintings in their home. The problem being that one needs a badge to access these items, or in the case of farm animals, one can also kill them.