The prime switch runs the electric fuel pump, nothing else.
It should only be needed after the generator has been sitting for a few days or longer.
A carburator bowl is like a toilet tank. It fills until the float shuts off the gas coming in with a valve.
While sitting, some gas can evaporate from the carburator bowl. Instead if cranking the engine to send power to the fuel pump, slowly filling the bowl, you bypass that with the prime switch. Once the carburator bowl is full, the fuel pump just pumps against the needle valve, like while running.
If its not starting quickly, after sitting only a few hours to a day or so, you may need the choke adjusted. That closes off the intake air, causing the engine to pull a bit more gas in. Once running, engine vacuum and a choke heater opens it.
If your handy, remove the air filter, and see if the choke valve is closed during startup.
Here is a great image of how carburators work.