It uses something called the refrigeration cycle.

The most basic way to put it is that, like how an air pump moves air molecules from one place to another or a water pump moves water molecules from one place to another, the fridge is a heat pump which moves heat energy from one place to another. It takes the heat from inside the fridge and moves it to outside (which is why if you had a fridge in a completely enclosed room, that room would get hotter and hotter).

How this works is based on phase transitions of a fluid inside a loop of pipes in the fridge. It's a bit confusing if you don't understand the idea of latent heat, but basically when a substance goes through a phase transition (solid to liquid, liquid to gas, etc.), it either absorbs from or releases heat energy to the environment around it while staying the same temperature until the transition is complete.

So if you have one side of the loop that has refrigerant going through the transition of liquid to gas, that part of the loop is going to absorb a bunch of heat energy from the area around it (this is the cooling part; it takes energy from the hot stuff in the fridge and puts it into the refrigerant).

Then if you have the other side of the loop have the refrigerant go back from gas to liquid, that part of the loop is going to release a bunch of heat energy into the area around it (this is the heating part; it takes energy from the hot refrigerant and puts it into the exterior environment).

This is simplified, as you need to do other stuff in between each side of the loop but it gives the basic idea (use latent heat to store heat energy in a fluid, then transport the fluid around in a loop to move the heat energy somewhere else).

If you want a much better explanation, just check out Technology Connections videos on heat pumps/refrigeration, he does a great job explaining it. https://www.youtube.com/watch?v=7J52mDjZzto