This device does no damage on its own, but it greatly improves the damage dealt by other Defensive Buildings near it.
It works like a Defensive Building Damage statue (albeit with a limited sphere of influence) and adds to the damage boost of the affected defenses, not multiplying it.
Keep in mind that, once the Damage Amplifier is destroyed either by troops or Gunboat Weaponry, the remaining buildings beside it will no longer be affected by the Damage Amplifier, thus reverting the defensive building's damage back to when it was not yet affected by the Damage Amplifier.
It is generally best to deal with the amplified cluster of defenses using Shock Bombs to stop the high amounts of damage that the amplifier will enable the defenses to do.
The range of the Damage Amplifier is relatively short, so it is best to place only Defensive Buildings near it.
Since the defenses must be clustered around the amplifier, try not to place several of the same type of defense near it since they will be vulnerable to being shocked together.
Two defenses placed on opposite sides of the amplifier each two tiles away from it will be far enough apart to be impossible to shock both with just one Shock Bomb.
At Mark I, the Damage Amplifier is a large blue box with a blue hatch on the front that sits on some wooden planks. It has two claws that are holding a small, light blue crystal. There is a length of wire running down one side.
At Mark II, the wooden planks are replaced with a metal base. The wire grows substantially longer and is coiled up. There is a smaller black box with a lightning symbol on one side of the Amplifier, which the wire connects to.
At Mark III, the claws turn golden.
This is the only way to improve your existing defenses' damage other than Statues.