The stator is like an alternator. It charges the battery while the engine is running (and does so better as the RPM's rise). The other electrical components will still work as long as they have power going to them, whether the stator is hooked up or not, they will work off the power generated by the engine, but they will drain the battery when the scooter's engine is turned off (if you for some reason leave the key in the "on" position.
I've had more than one alternator go on my various cars, and it's the same thing. The car will start with a jump or with a fully recharged battery, but the battery will not charge and a jump or recharge will be needed after a day or two.
2009 Kymco People 150
1983 Honda Sabre v45 750
2006 Yamaha Vino Classic 50 (no longer owned)