Hi Popeyemth,
Yes, it does transfer the load automatically. However that may create problems over time.
If the circuit is heavily loaded with say a 1500 watt electric heater, when it transfers it may "burn" the contacts a bit each time. Eventually they may fail.
It is really best if the transfer happens when there is little or no load. In my opinion that kind of defeats the purpose of an automatic transfer switch.
In this particular case, because the fridge is likely a low load--the transfer switch may be just fine.
The fridge, on the other hand, unless the transfer is very quick, may not like having a "blip" in the power supply when it is actively cooling. One way around that is to have a transfer switch that includes a three minute time delay. Another is to have a transfer switch that reacts in the millisecond range.
In case you have not yet guessed--I'm not really much of a fan of automatic transfer switches.
In the OP's situation I'd simply run the fridge from the inverter at all times except when the RV is in powered down storage.