To unlock the Master Exploder trophy in Borderlands: Game of the Year Edition, you will need to kill 25 enemies with explosive weapons.
Explosive weapons can be identified by the yellow explosion symbol in your inventory (similar to the trophy art image). You can also do explosive damage by shooting the yellow barrels or spawning a vehicle with a rocket launcher from the CATCH-A-RIDE.
When you use an explosive weapon, yellow numbers will appear around an enemy to indicate that you are causing explosive damage.