Fridges work by taking the heat out of the stuff/space inside, and radiating it out the back, effectively. They aren't perfectly efficient, so if you managed to keep the fridge working while the door was open all that would happen would be the room would just get hotter... They don't just magically make cold.
When water changes to water vapour, or ice to water, it takes heat out of its surroundings, so it's not just the low temperature that cools the air if you blow a fan over it, it's the energy it takes (in the form of heat, out of the air) to change phase.
Fridges work by taking the heat out of the stuff/space inside, and radiating it out the back, effectively. They aren't perfectly efficient, so if you managed to keep the fridge working while the door was open all that would happen would be the room would just get hotter... They don't just magically make cold.
When water changes to water vapour, or ice to water, it takes heat out of its surroundings, so it's not just the low temperature that cools the air if you blow a fan over it, it's the energy it takes (in the form of heat, out of the air) to change phase.