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You can read a lot about this by looking at the Stereophile preamp measurements.
Take a look at the frequency response into various loads. That's the core of the issue. We talk about output impedance and input impedance, but what it boils down to is if these are mismatched there is reduced output. Sometimes at some frequencies, sometimes all across the board.
Another thing to look at is the STereophile amp measurements for tube amps. Look at the "simulated speaker load" graph.
Was I expected to read the whole post? Why? The thread title is all the question needed.
Input impedance is like a pressure the other component has to work against. The other component can be a cartridge going into a phono stage, phono stage going into a preamp, preamp going into a amp, or amp going into a speaker. It is all the same concept.
If the receiving device has a low impedance, low resistance, then the component driving it will need to be able to deliver a lot of volume, aka amperage, aka current flow, to drive it. Or else it will "run out of gas" and you will hear this as a loss of musical drive and energy, and often times worse bass response. If the driving component is weak in terms of power supply then impedance matching becomes ever more important.
There is also another way in which impedance matching matters, and it works in a way that really complicates matters and confuses a lot of people. Send a current down a wire and it goes and goes pretty much the same the whole way. This we glorify and call transmission line theory. It goes and goes just fine, until it encounters a disturbance, usually a connection or the next component. At this point if the impedance changes a lot this sends a sort of shock wave back along the wire all the way to the source. Where if it again encounters another impedance change the shock wave bounces back again. This all happens very fast and produces a sound we call ringing.
Ringing happens most notably in MC phono cartridges where the high impedance is great. Because the phono cartridge cannot put out much current. But too high and the ringing accentuates the top end. So we "load it down" by lowering impedance until we get the sound we like.
Unfortunately something we cannot do with amps and preamps, thus these rules of thumb they keep coming up with, because it is a whole lot easier to tell someone a rule they don't understand than to explain it like I just did so they do.
Most aspects of the sounds of music reproduction are subjective because there are so many variables involved and what people want is variable. You can spend lots of time with measurements and get no where. While there are some variables sampled in the specs that can tell you the big picture (inefficient speakers and flea sized amps don’t work) they don’t tell enough of the story to allow good component or compatibility choices... Any one variable in itself (unless completely anomalous with standard audio characteristic) is virtually irrelevant in getting the sound you want.
I started into high end audio when I was a working scientist. I quickly learned to just check the top few variables and then ignore the specs for choosing components.. They can tell you a bit about the component... like the power... but there are so many not measured that you just can’t really assess the sound and compatibility with them. Even as someone that loves data, charts, and relationships... I don’t look at them. To me, a waste of my time. Some people really like them... but at best they can only be the very faintest of indicator.
You have to listen to them yourself and read professional reviews of the same units. This way you calibrate yourself to the generally accepted characteristics in sound and terminology, learn what the unit sounds like to you, and also realize what sound characteristics you like and want.
All reviews are of systems. So within all reviews are information about the system and its components. Usually a reviewer reviewing a preamp will use several amps to check for compatibility... and document the differences. There are lots of high level generalities... but they are just that... generalities and any unit can be vary different. Over many years and decades if you work hard and are good with unstructured problems you will develop the ability to read reviews and know “house sounds” well enough to put together the system you want.
This is a great challenge and pleasure to some, something one can be fascinated for a lifetime and reap great rewards. Or it can be incredibly frustrating if you are not good at it. If you are the latter, then you either throw up your hands and say it’s all bs or you go to showrooms, listen to systems and have someone piece together something you like. I have an incredible system that brings incredible enjoyment to me that I have built over fifty years. My best friend has a $30K system that sounds absolutely terrible.. he was a scientist as well he would read reviews and choose award winning components and when he put them together the sounded absolutely terrible. His system has sat there untouched for ten years.