How does the bandwidth have something to do with 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.