They are saying that, if the bandwidth of a certain device becomes narrow, then its speed decreases. However, how does the speed has something to do with the bandwidth? If you are saying that the bandwidth is located at very high frequency, then perhaps if you shift the bandwidth to lower frequency, then the speed decreases. So, how does that has something to do for speed?
This is not hard to understand if you grasp the difference between carrier frequency and signal.
The way information is transferred over long distances is with carrier frequencies that are modified just a tiny little bit. A carrier frequency is nothing but a sine wave at some specific frequency; by itself it contains no information whatsoever, because you cannot get information from a sine wave that is always the same.
The information comes from changes that you make to the carrier signal. For instance, you can make a rule that changing the carrier by 1Hz represents a one and by -1Hz represents a 0. Or maybe that lowering and raising the amplitude conveys information. Or one of many other coding schemes.
What is important to understand is that it is the modification of the carrier that conveys information. And this is not something you can do with infinite precision; both the transmitters and receivers are subject to interference or noise that make it impossible to distinguish between very small changes in the carrier. So you always need a finite amount of space around the carrier frequency to convey a nonzero amount of information. This is called the bandwidth, and only it determines your speed. The carrier frequency is not important.
The carrier frequency is important for the total amount of information that can be fit into a certain range of available frequencies. You don't get all the spectrum in the world to work with. For one; higher frequencies can't penetrate large obstacles so they are unsuitable for long-range communication. Some frequencies are also quickly absorbed by the atmosphere. This leaves only small pockets of bandwidth (e.g. 2.4-2.5GHz for ISM, the stuff that old WiFi works on) that are usable and even those need to be subdivided into 'channels' (e.g. 2.40-2.405GHz) to make sure that multiple adjacent transmitters can work without interfering with each other.
Regarding the time domain behaviour of a system with limited bandwidth (in particular: filters) there are two effects:
1.) The switch-on transients (until steady-state conditions are reached) is inversely proportional to bandwidth: Smaller bandwidth causes larger transients. That means that the time until the desired signal can path the bandpass is larger.
2.) Group delay: This parameter is the most important property of a frequency-selective four-pole as far as "speed" is concerned. Group delay is defined as the negative derivative of the phase response. Because the bandwidth is inversely proportional to the slope of the phase response (small bandwidth with large phase slope) we have a large group delay for a small bandwidth. With other words: A low group velocity for a smaller bandwidth.
(Comment to "group velocity": This is not always identical to the "speed of a signal" - it is the speed of a group of frequencies having a bandwidth which is small if compared to the mean value of these frequencies).
Does this (at least partly) answer your question?