You are confusing watts with amps.
The amps (for a given load) is ten times as much at 12 volts as it is at 120 volts.
Example:
- watts at 120 volts is 1 amp.
- watts at 12 volts is 10 amps.
watts is Volts times amps
The inverter is putting out 1 amp, because it is at 120 volts. The battery needs to supply 10 amps to the inverter because it is only at 12 volts. Power into the inverter is (almost) power out of the inverter.
The inverter just converts low volts DC to higher volts AC.