The electric part of an RV refrigerator is a small heater of about 300 watts which is inserted into a socket in the refrigerator chimney. If you look at where the propane flame is there will be a (usually circular) metal pipe like enclosure, rising straight up from the flame. On the side of that, you will find two wires going into the "chimney". If you have a 12/120 volt unit, there may be four wires.
The wires go to the heater(s), always 120 volts and maybe 12 volts if you have that option. The smaller wires go to the 120 volt heater. Disconnect them and measure to see that, when on 120 volts, you have power to them. Then measure the resistance of the heater wires themselves. You should measure a fairly low resistance across the 300 watt heating element.
If you have 120 volts to the element, but high resistance across the element, you have a burned out heater. It can be removed from it's socket in the chimney and replaced. The element itself is usually under $50. Be aware that sometimes the element is rusted in place (it simply slides into the pocket) and may not come out easily. Also, if it has burned out, sometimes it is welded in place, again making it very hard to remove. However, I have had little trouble replacing the 4 or 5 I have done over the years. Try removing it prior to buying a new one at an RV dealer. If it simply will not come out, just use propane all the time for your refrigerator.
That should get you started.