The inverter receives 12VDC from the batteries and puts out 120VAC.
On some models there is an internal transfer switch to pass the 120VAC Shore Power straight through the inverter to the receptacles and provide 12VDC battery charging. These units only pull power from the batteries when you do not have shore power.
You need to go back to the batteries and make sure there are no corroded connections and all connections are good and tight. Loose or corroded battery connections will let the voltage go through but not enough amps to power anything.