Your "inverter" is running all the time (if you have it on). There is a transfer switch that will switch the AC power, either generated from the battery OR from shore power (AC). Once you unplug your camper from shore power, the transfer switch will switch over automatically and then the AC power being produced from the inverter will power things in the camper. The inverter gets it's power from the battery.
To simplify: Your solar charges the battery. The battery powers the inverter. The inverter transfers battery power (DC) to AC power (house hold power).
When you are plugged into shore power: Shore power passes through the transfer switch and the transfer switch shuts off the power coming from the "inverter" because you are plugged into shore power and don't need to run things off the battery.
You battery is charged by the solar system. Your battery is also charged when you are plugged into shore power because the "converter" in your camper changes AC power to DC power and then charges the battery.
Again, when plugged into shore power, the transfer switch will keep the main AC power coming through shore power, not the battery.
It's kind of a circle when you think about it.
I hope this makes sense and isn't confusing you too much.
So, basically, the inverter is in use when you have no shore power connected to the camper. The inverter can be turned on and off. But I think most folks leave their inverters running all the time so when breaking camp or traveling, they don't have to think about flipping it on so they have AC power running in their campers (especially residential refrigerators).
And, if you ARE connected to shore power and if you have power failure, the inverter will automatically switch over to your battery and keep the AC flowing without interruption.