The AC power (household 110 Volt power) is produced by the inverter from the DC power in the batteries. Here are a few things to check if there is no AC from the inverter.
The battery charge circuit includes the batteries, solar panels, charge controller, 20 Amp circuit breaker, and the wiring that connects all of them. Problems with any one of these can interrupt the current flow into the batteries.
The portable solar units are very reliable and normally require little maintenance, but please take some time to familiarize yourself with the ebb and flow of solar power --- it will help greatly when something does go wrong.
Grover--- Resident prognosticater. If you need specifics, contact me at firstname.lastname@example.org.