Good question! I will assume that you manage to trap pure steam in the container (no air is left in the container), and that the container is completely airtight with no leaks. I also will assume the container is rigid and won't crumple up when the pressure inside drops!

Here's the simple answer:
When you put the container in the fridge, the steam, which was over 100°C cools down. This means it will try and condense into liquid water. However, as some droplets of liquid water start to form, there will be less gas in the container and so the pressure in the container will drop. When the pressure in the container drops, the boiling point of water will also drop. If too much steam condenses into water, then the boiling point of water will drop below fridge temperature (lets say 4°C), and some water will turn back into steam, raising the pressure of the container. Then, if the pressure is higher, the steam will want to condense again... and so on, and so on.

The result of this process (which happens very quickly), is that a steady equilibrium will be established between the liquid water and the water vapor (steam). Most of the steam will have turned into water, but just enough of the steam will remain as a gas so that the pressure in the container is high enough to keep the boiling point of the water right at your refrigerator temperature. This equilibrium pressure will be about 1% of the atmosphere pressure (see below).

Opening the container will be difficult due to the low pressure. But, if you manage to get it open, then air from outside will immediately rush in. The pressure will rise up to atmospheric pressure, and the boiling point of water will go back up to 100°C. Since all of your remaining water vapor is refrigerator temperature, pretty much all of it will immediately condense into liquid, with maybe a little entering the air.

More details:
If you want to know more, you can use the ( phase diagram of water ). This tells you (among other things), how the boiling point of water (line between "Liquid" and "Vapor") changes with temperature and pressure. At 1 atmosphere (10,000 Pa or 1 bar), the boiling point is at 100°C. But, at 10 mbar (0.01 atmospheres), the boiling point drops to about 4°C, which is refrigerator temperature. This is how we know that the pressure in the sealed container will be 1% of the atmospheric pressure when the container reaches equilibrium in the fridge. You can go further and use equations like P V = n R T to figure out what percentage of water molecules will end up in the liquid vs vapor at equilibrium- I got that about 1.3% of water molecules will end up in the vapor phase, and the rest will be liquid water.

There are four phases of matter that we are currently aware of: solid, liquid, gas, and plasma. What state something is in depends on its:
pressure (P), temperature (T), composition (X), and oxygen fugacity (fO₂). Let's take a look at your question from a thermodynamic perspective.

I've put a phase diagram (a graph that depicts the different states of matter according to varying P-T-X-fO₂ conditions) for water below for you to follow along with.

We are at 1 atm pressure, so first find where that is on the y-axis. Steam is the gaseous phase of water, and if P = 1 atm, we can see from the phase diagram that the temperature of the steam must be between 100 degrees C and 374 degrees C (Tcrit for H₂O). We know this because there is a boundary line there (we call this a phase equilibria line), along which two states (in this case, gas + liquid) coexist in perfect thermodynamic equilibrium. So, at the point where you have collected the steam, we must be in the regime of the P-T diagram that I've indicated using the green rectangle. Once you put the container in the fridge, the temperature of the container will decrease (in the direction of the black arrow) until it comes to thermal equilibrium with the air inside of the fridge (usually set ~2 degrees C). How long this takes is a function of many different parameters (you can look up Fourier's law of thermal conduction if you're really interested). I'd guess that two hours is more than long enough for the container to cool to a lower T, making a phase transition from gas to liquid. You'd likely find water at some temperature between 0 C and 100 C (I've outlined it in the blue rectangle.)

I hope this has really helped to explain WHY phase transitions happen. Thermodynamics is some pretty cool stuff.

The steam collected from the boiling water is going to be the pure gas form of the water boiled. The water molecules are really hot, so they are moving around really fast, which is how they escaped the surface of the water into the air. Once they’re cooled in the fridge, they’ll condense back into liquid form and you’ll find a little cup of purified water!

You would find dew all over the inside of the pot. This is because water that evaporated from the boiling water has condensed on the inside when it became cooled off by the refrigerator.

That sounds like a good experiment for you to do. How would you collect the steam? The steam would turn back to water after the temperature dropped below the boiling point.