Why do OOK transmissions have bandwidth?
Only an ideal sine wave with constant amplitude and single frequency that runs for infinitely long in time has extremely narrow bandwidth of a single peak. Having any changes to the ideal sine wave such as changing the amplitude means that since it cannot be represented as a single sine wave any more but a sum of many sine waves of different amplitudes and different frequencies, in frequency domain it means it has a range of frequencies. The faster the changes, the wider the bandwidth.