Both the fridge and lights are powered by 12V DC. The converter supplies the 12V when connected to shore power.
There may have been an excessive load to the converter when you first connected to shore power, tripping a breaker or a GFCI. If you have tripped that GFCI a number of times, it will need replacement. If your battery is discharged excessively a number of times, you will need to replace the battery.
It would be helpful if you stated what trailer you have and describe the electrical portion. I have no idea if you have a horse trailer or a fifth wheel.