This is a fable. When a fluorescent light is first switched on, there is a higher current draw for a second or so. However, the length of time for this higher current draw is so short that it has almost no effect on energy use. Leaving a fluorescent light on for more than a few minutes will use more energy than turning it off and turning it back on when needed.
A ballast regulates current to the lamp during startup and operation. Turning lights on and off frequently will have some impact on lamp and ballast life. However, the old principle of leaving lights on if the room is unoccupied for more than 15 minutes no longer holds true.
Lighting manufacturers now recommend that fluorescent lamps be turned off if the room will be unoccupied for more than two or three minutes. This is the result of technology advances in electronic ballasts, which can handle frequent on-off cycling with little loss of lamp and ballast life. Fluorescent lamps with instant-start ballasts are the exception, however. These apply a higher voltage during startup that increases efficiency but may reduce lamp life. These lights should be left on longer (up to 30 minutes or so), to minimize the potential impact of more rapid on/off cycling.