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.

On-Off Keying (OOK) is equivalent to AM modulation of the carrier with the data bit-stream with a 100 % modulation index.

As a result the entire spectrum of the bit stream is up-converted to the carrier frequency of the OOK modulator. Additionally, due to AM modulation an image of the data-stream exists below the carrier frequency in addition to the translated spectrum above the carrier.

The bandwidth is not due to carrier frequency drift. As a first approximation, the bandwidth is twice the frequency that modulates the carrier:

This may not be intuitive, but consider that the amplitude modulation (including OOK) is essentially a multiplication of carrier and modulation signals, and basic trigonometry tells us that

$$2 sinA sinB = cos(A - B) - cos(A + B)$$