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It changes the way the pedal responds to your inputs. Generally, they make the throttle more sensitive. Admittedly, the F's standard throttle response is pretty slow even compared to the IS350, but it all disappears when you press SPORT. The aftermarket controllers give you more options by changing the signal coming from the accelerator pedal.
It will not change the engine's power, so what's really happening relates to how the car feels relative to your foot action. Some people like it.