Exploding Barrels are obtainable through crafting, Black Market Goods (stolen from mobs in dungeons), and other rewards. The recipe is obtainable through killing bosses, as well as purchasable with Shrunken Crystals of Britannia.
They can be used to blow apart Tentacle Vines found in the overworld, which leave behind a Bag of Loot on the ground.
They can also blow open gates in dungeons to get access to a locked chest containing loot (unknown at this time).
They also can do significant damage to mobs in the explosion radius.
To use them, place them on the ground, and double click them.
Blaze Catalysts are dropped from mobs in Despise, and are used to create an “Enhanced Exploding Barrel”, which is used to blow open the gate to the Despise boss.
Fuses can be used to set a timer on an exploding barrel, delaying their detonation. This is crucial for the Enhanced Exploding Barrel, as the explosion radius is so large that without a fuse the player using the barrel will kill themselves.