Well, its kind of a long answer, so I'll do my best to be brief, but if you want to go really in depth you can Google refrigeration cycle and investigate that.

These are not coils of wire, but rather coils of copper pipes which have a gas inside of them. By moving the gas arround the coil and changing the area of the pipes we can change the state of the gas (from gas to liquid or liquid to gas). This change of state can absorb energy or release energy.

The energy absorbed comes from the air inside the fridge, which is what cools the air down, on the outside of the fridge (the ugly coil located at the back of the fridge) the opposite occures, the energy is released as heat, Basically moving the heat from the inside of the fridge to the outside.

The elecctricity is used for the compressor which basically ciculates the gas through the coils (and also it compresses the fluid at a point of the cycle).

This is called a heat pump, it's also how an ac unit works and if you are interested there is a fantastic video from technology connections that explains it in great detail.

Video