Quest
The bomb is an item associated with the quest to get off Plunder Island. Part of this quest requires finding out who Deep Gut is and where he is. A reporter has this information but refuses to share it unless you perform traditional pirate tasks to give him a story to write about. The second deed is to become the captain of a ship known as the Screaming Narwhal. The bomb is needed to complete this deed.
Location
The bomb is in the jungle of Plunder Island. You enter this area from the "Jungle Entrance" on the island map. Once in the jungle, you need to head west. You eventually find a pile of bombs on the floor. You can get one of the bombs by examining the pile.
Grease Fire
The bomb cannot be used on its own. You need to create a grease fire before you can use it. The grease spills automatically when you attempt to walk up the gangplank of the Screaming Narwhal. This creates a puddle of grease on the dock that can be lit. In order to light the grease, you need to attempt to climb the anchor of the ship. Captain Winslow throws burning coals at you when you do so. These coals set the grease on fire.
Using the Bomb
Use the bomb from your inventory on the grease fire to light it. Run to the clothesline connecting the ship to the dock. Use the lit bomb on the frilly underwear to place it inside. You need to attempt to climb the clothesline at this point. Captain Winslow attempts to rotate the clothesline to move you away from the ship. This causes the lit bomb inside the frilly underwear to move toward him. The explosion knocks Captain Winslow off the ship, which results in you becoming the captain of the Screaming Narwhal.