When a fluid evaporates, it removes heat. If you want to try for yourself, you can lick the back of your hand and blow on it to dry it. It cools down, and a fridge uses the same principle.
Inside the fridge there is a long coil containing a coolant. The coolant boils at a very low temperature, and even if it is already fairly cold inside the fridge, there is enough heat to make the coolant evaporate. When the coolant evaporates, it absorbs heat and therefore, the coil inside the fridge feels cold. The coil containing the evaporated coolant continues outside the fridge into a compressor.
The compressor is a kind of pump that pressurizes the vapors and pushes it into another long coil called a condenser. During its journey through this coil, the vapor condenses and becomes liquid. This is the exact opposite of what happened inside the fridge. When the coolant condenses from vapor to its liquid form it dissipates all the heat it previously absorbed in the process of evaporation. Therefore, the coil outside the fridge feels warm.
In other words, the coolant absorbs heat inside the fridge and releases heat outside - and then it flows back into the fridge and starts all over. It is the compressor that keeps this circuit up and running. It does so by creating a high pressure in the part of the coil that is located outside the fridge - and it is this pressure that makes the vapor condense and dissipate the heat. Only when the coolant has become a fluid can it pass through the thin coil and return to the fridge. In there, the pressure is so low that the coolant can evaporate. When you turn the handle, the pressure inside the fridge changes and makes the coolant circulate.
For the boffins
What's the little pipe in the middle of the cooling system doing?
The capillary pipe serves to keep the two sections of the cooling system apart in order that the pressure difference is maintained. The capillary pipe is so thin that the coolant can only pass through when it is liquid. In other words, it makes sure that the coolant is not allowed into that part of the cooling system from which it is supposed to absorb heat until it has cooled down enough (to its liquid form) to actually start absorbing heat again.