Generally they have a capacitor power supply, with a triac providing the drive. They get power when on by only switching the triac on after the instantaneous voltage is over around 20V, so they can have an internal regulated rail.
The problem with them and energy saving lamps is the low power draw when off is often enough to charge up the internal capacitors in the lamps, and this causes the lamp to flash every few seconds ( a single short dim flash normally) as the internal lamp circuit starts up and discharges the capacitor. This kills the CFL unit quite fast from the repeated failed starts.