Connect the black wire coming out of the inverter to the positive (+) terminal of the battery. Most inverters allow you to do so using alligator clips.
Connect the red wire coming out of the inverter to the negative (-) terminal of the battery.
Plug an AC-powered appliance into the inverter. The inverter will still charge if you are using it concurrently; it will just do so more slowly.
Wait for the inverter to charge. This generally takes one to two hours, but it varies by device. Check your manual for the exact time.
Disconnect the cables in the reverse order -- red first, then black. Your inverter will now be charged and ready to use without the battery. When it runs out of power, just charge it again using the same procedure.