Why Power Cables Affect Sound

It could be technically correct that the power cable is not in the signal chain but it can limit and alter the amps ability to accurately reproduce the signal by messing with the power. 

It's not that hard of a concept to wrap one's head around. I've experienced a limiting of the soundstage, dynamics and a general softening of the music, obscuring details, ambience and extension as a result. All of that by use of a lesser power cord than what the situation demands. 

Not in the way that a speaker cable or interconnect can do but in the end, the results were strikingly similar.

All the best,
Nonoise

Other factors related to all power cords affecting sound quality, including tricks of the trade,

1. Type of wall outlet
2. Type of wall outlet cover
3. Elevating or suspending power cord
4. Demagnetizing power cord
5. Ionizing or otherwise removing electric static charge
6. Using a contact enhancer on all electrical contacts
7. Choosing a white cable jacket
8. Judging sound quality only after a burn in period of at least 200 hours unless using a burn in device

No, the AC cord does not carry the audio signal, but it does carry the electrical energy upon which the audio signal is delivered throughout the system and to the speakers. How free of noise and distortion for the audio signal is that energy? It is often said that amp transformers are an extension of the power cord. Is the noisy AC cleaned up in the transformers, or is there a continuation of a burden of distortion that the audio signal must share, resulting in sound degradation that you hear?
Why do you install aftermarket caps, crossovers, hook-up wire that are better in materials than original---to improve clarity, I presume? I don’t do these things, but I have addressed AC power and AC cords, vibration, EMI---all changes have brought vast improvements in clarity and increased information.
A lot of folks want scientific explanations. Others have had their minds made up for them by authorities---no need to try it for themselves. I regard these positions as obstructive to progress. There is no technical explanation that will satisfy them---just read the hyperbole from a cable manufacturer--still not good enough. For others, there is no room for authentic knowledge--the certainty gained through your own experience. IME, the electrical energy has to made clean and then kept clean. I didn’t know anything---I listened to others and then tried it---successfully. As "NoNoise" just said, not that difficult to wrap your head around. 
Power cords help remove the noise and distortions from the energy that delivers your signal. Is a six-foot cord enough to do that? In many cases, no, and that is why folks don’t hear enough of a difference. The power ahead of the cord is still too noisy and not good enough for high-performance audio. Dedicated circuits give more power, but not always the clean power needed. A power cleaning appliance, made by, e.g., Equitech, Furman, Richard Gray, Torus, PS Audio, et al, is what is really needed to ensure power that is free of distortion and noise. Your nice cords usually plug into that appliance or shortly thereafter.

kosst, I make points often, that not everything will show up in measurements, such as this power cord thing. Comparing 2 12 gauge pcs from different manufactures, I hear differences. I have built cables, using the same wire, but different connectors, and I hear differences. The opposite is true, too, same connectors but different wire. Before I determine which is better ( which would be up to me anyway ), I listen for the differences, and they do exist (just like the M&Ms guy and Santa, in the commercial, lol ? I am not sure measurements can reflect these differences. I changed similar values of storage capacitors in an amplifier recently, and I heard differences, and the amp was not that old where the caps needed to be changed. I spent money to upgrade. Before I ever determined which capacitors I preferred, I concentrated on listening to the differences, with my test music. And then there was the " break in " period, which we disagree on as well. I do not know if these capacitors would show differences when being measured. Measurements are necessary, but, in the end, it is listening that's matters most. Peace and joy to all.  Enjoy ! MrD.