A heatpump works on basic refridgeration principals. It uses electricity to run a compressor which pumps refridgerant gas around a system of copper pipe, thus the reduced power consumption.

This refridgerant gas acts like a sponge for heat. When it is expanded in a vacuum it soaks up heat from the atmosphere that is conducted through the copper piping. When it is compressed it does the opposite it gives off heat into the atmosphere by being conducted though the copper piping.

Therefore a room can be heated by expanding the refridgerant gas in the outside unit therefore soaking up heat, pumping that refridgerant through to the indoor unit and then compressing it therefore giving off the heat.

A room can be cooled by reversing the flow of the refridgerant gas therefore soaking up heat from the indoor unit and expelling that heat out at the outside unit.

While the indoor unit is cooling the metal plate contained in it gets cold, this then gets condensation forming on it. This condensation is actually moisture out of the air in the house. The condensation quickly forms into water droplets and then flows down the interior drain in the indoor unit and out side. This means your house is being dehumidified.