OK I can give you the easiest way to logically understand what is going on. So lets do a thought experiment in the huge arena of time-invariant linear systems, and the basic idea of stimulus/response (also called input/output). Physicists and especially engineers analyze a huge category of problems assuming they can be analyzed with what are called linear differential equations. Engineers go to great lengths to design systems around the advantageous properties of linearity which among the advantages means that sine waves and cosine waves** when applied (as stimulus) to a linear system, generate in the response no additional sine waves of frequencies not present in the stimulus. So now that we establish the importance of linearity and sine waves, let's consider non-linear systems. (BTW the propagation of electromagnetism in free space, and in electrical and optical materials within certain power limits, is a linear system process thank Goodness.)
In communications, all modulation techniques (modulators) are non-linear, non time-invariant which (by the previously mentioned properties of linearity) means that with a constant amplitude sine wave*** applied to an example of such, a multitude of different sine wave frequencies result in its response (which we call its 'output'). The modulator output is what we call a modulated signal. This is the case for all means of electrical communications including optical, as the only differences between say coaxial cable, microwave links, and optical fiber is the matter of where in the electromagnetic spectrum the range of each lies. All of the information transferred is made up of the patterns of diverse sinusoids (a general term for sine and cosine functions) present in the signal. The more information per second (we say bits/sec) the more spread out this multiplicity of frequencies is.
Now suppose a hypothetical channel is ideally the narrowest bandwidth possible say zero BW, and you apply a signal that has been modulated to this channel and the channel straddles the original unmodulated sine wave signal. Now even though a channel like this is not possible even in theory, we can pretend it exists and say the channel is an "infinitesimally narrow passband filter" in engineering parlance and we apply the modulated signal to its input.
So this infinitesimally narrow channel would have, in its output, a strange effect on the modulated signal, as would any virtually zero BW or extremely narrow channel, by the properties of linear systems. The transferred signal would be stripped of its modulation and so the channel output would be the original constant amplitude sine wave applied to the modulator, in which the only information present would be a scalar value for its phase. No other data would be present. The longer this sine wave would be measured and analyzed, the effort would only increase the accuracy with which the scalar value would be established, based on information theory but that is it - the channel is useless for anything else.
So by this extreme example of showing why zero BW is not tolerable I hope to have shown why progressively larger BW would let through progressively more information per second or bits/sec. And this is actually quantified by the famous Shannon-Hartley theorem which is the governing principle for all types of coding/modulation, of which the questioner's on/off states (on/off keying) is the simplest and is not particularly bandwidth efficient. So it cannot be intuited by the maximum frequency possible of on/off states (or the related amplitude keying), whether optical or electrical. This last pitfall is something students (myself included, way back) are tempted to employ to grasp what is going on here but it is a pitfall.
** ~ what are called 'eigenfunctions' of linear systems ~
*** If it were varying amplitude, it would not mathematically be a sine wave.