Think of the KB like your intake plenum, except now it pressurizes the air. Now think of the air-to-water intercooler like part of the lower intake manifold, except it has passages for coolant to flow through, just like your engine has passages for coolant to flow.
When the supercharger pressurizes the air, the temperature goes up. It then passes from the supercharger through the intercooler. The metal of the intercooler draws heat from the air, and the coolant flowing through the intercooler draws heat from the metal. The heated coolant then flows through the heat exchanger, which removes heat from the coolant. The coolant is then pumped back into the intercooler, and continues on this cycle.
This isn't an intercooler like you see mounted on the front of cars. This intercooler is basically a block of metal mounted between the engine and the supercharger. The supercharger sits on it. The heat exchanger is what hangs in the front of your car and uses air flow for cooling. This cooling is then transferred to the intercooler via the coolant.
This coolant is separate from your engine coolant/radiator system, hense the new reservior. It even has its own pump.