Hi,
The LED acts sort of like an RC circuit, where the 'equivalent' capacitance slows down the turn on and turn off times. This is mostly because the LED would be driven by a current source or something like that. But if R can be kept low and we allow a higher than usual current to flow into and out of the LED, we can get it to switch faster because that means we can control the carriers into and out of the active region faster and that's what turns it on and off.
So what to do would be to apply a fast rise time voltage pulse to the LED and allow the current during the transition to go much higher than normal, and a fast reverse voltage pulse to turn it off.
Some LED models should not be pulsed too fast though as the manufacturer will only recommend low frequency pulsing for some types like 1kHz to maybe 10kHz. Consult the data sheets.