AC means alternating current which is what you get when plugged in. Some fridges operate on gas or AC. Some older ones operate on gas, AC or DC which comes from your battery.
The fridge is not like the one in your house. You have an absorption system that requires a closed circuit in which refrigerant circulates. The circulation is caused by heat which is where the flame comes in. If your fridge requires the lighting of the pilot lite, then this is it. This is all the flame you need. It does not "come on" like a water heater does.
Now, you also need patience. Once the pilot is lit, it might take hours to get things to cool down. Depending on the ambient temperature outside, it might take a day. If it never cools down then you have a different issue.