Think of it like this:
With the exception of the air conditioner, microwave, part of the refrigerator, and perhaps part of the water heater all the appliances in the trailer use 12 volts DC.
When not connected to a 120 volt AC source:
The 12 volts is supplied by a battery (or more).
One just has to live without the AC and microwave.
The water heater and refer use propane as a fuel, but need the 12 volts to power the control boards, or brains.
The lighting, furnace(while also burning propane), and water pump run from the voltage supplied by the batteries.
When connected to a 120 volt AC supply the following happens:
The ac and microwave will now function.
The convertor changes the 120 volt AC power to 12 volt
DC, which is used to recharge the battery and mostly power the 12 volt appliances.
The refer's 12 volt control board sees the 120 volts ac and turns off the propane burner, activates a 120 volt heating element, and goes about keeping the beer cold.
Your trailer may or may not have a 120 volt option for the water heater. If so, there are likely two switches for control. One is a small 12 volt switch with a red light next to it. The other is usually a normal 120 volt light switch. You chose which system is heating water by which switch is on.
Hope that helps.