Here is Louis' longer version. Some people say, "why worry about power quality using all sorts of filters and specialty power cords when we can simply use batteries and bypass the problems of mains pollution altogether?" To which the experienced audiophile will reply: "Because the sound quality from batteries depends on the quality of power which was used to charge them." This sounds heretical since electricity is supposed to be a mere commodity, meaning that wherever it is derived from, it represents nothing more than the same simple potential (pressure) difference of negative charge. This turns out to be too simplistic and does not account for something which hitherto has not been explained by measurement gear - the concept of power quality

At LessLoss, we don't pretend to be the sole bearers of knowledge with regards to what exactly makes up all power quality parameters. But we have carried batteries around and have charged them under so many different conditions as to begin to glean patterns and predictable behaviour from what is supposed to not matter. Quality of power does matter even there. In some yet unknown way, there is a sort of memory involved. This does not yield to simple measurement but can be sensed, repeated and relied on by experience. The quality of power, whatever it is, carries over into sound quality. When we think about power being formed by a moving magnetic field along a wire—all power plants do this except solar cells—and sound from a speaker being formed by exactly the same process in reverse (moving electrons within a stationary magnetic field cause the vibration of the wires if the magnet is not allowed to move), one could say that the generation of movement and the cause of movement are the same in either case, just viewed from a different vantage point. And indeed, every motor can be cranked to become a generator and every generator can be driven by electricity to create movement. All we do is convert one form of energy to another.

Gryphon Audio Diablo 300 with LessLoss Firewall

When we hear sound differences due to power conditioning nuances, think of it this way as well: when we condition the speaker driver in some way, we also condition the resulting sound. Let's say we change the number of windings on the voice coil. The sound will change even if the signal given is identical. Let's change the magnet strength. The sound will change. Let's change the gap between coil and magnet. The sound again changes. Let's change the magnet type. Again the sound changes. Let's change anything in the motor (every driver is a motor). The sound will always change. So we have now established that we are incredibly sensitive listeners. Let's leave the speaker driver alone and change some quality of the power. The sound changes. Why? Conventional explanation has it that there are several factors we can change in the power supply, all of which influence sonics. 

1/ High-frequency noise. This creates intermodulation distortion in the source and amp and we will hear its artificial components superimposed upon the 'clean' signal. I wrote a paper about this here and an additional article here and also here.
2/ Low-frequency conductivity. This affects the damping factor of the amp. Energy swings from the speaker can to some extent 'control the amp' instead of the amp controlling the speaker when power impulses are restricted. We can hear this when we switch from a very thick wire to a very thin wire.
3/ Contact pressure. If we have loose power contacts, we hear a lack of control in the audio. Again we are losing damping factor in the amp.  
4/ Metal platings. Our plugs and sockets influence sonics due to their surface condition. Oxidation results in a muddled type of sound. Uneven contacts can restrict power impulses, leading to loss of control of the speaker driver. 
5/ Vibrations. Related to 3/, micro vibrations (or in the case of transformers, sometimes clearly audible bold buzzing vibrations), have an effect on the stability of the sound image. 
6/ Capacitor materials, capacitor vibrations, capacitor electrostatics all have a direct influence over the stability of the sound because the power values are influenced and changed by such factors. 

Gryphon Audio Pantheon speakers.

In general you see a tendency. It is possible to look at all of these minute power supply changes exactly as you would consider minute changes in the magnet of the speaker. As a thought experiment, it is easy to imagine that there is indeed a way to subtly change any given magnet in any given speaker to elicit exactly the same sonic changes heard when any item on the above list from 1 to 6 occurs. Just as the perfect speaker driver would have no ringing or movement of the magnet in relation to the voice coil, so too would perfect power pass all circuits on the way to the speaker in a way as to create perfect audio. You can create a loose power contact or a loose magnetic flux. You can change a metal plating or the chemical composition of the magnet. You can add high-frequency noise or introduce it in the form of a noise-laden active magnet coil instead of a permanent magnet. And finally, you can lower the conductivity of your power line or lower the efficiency of the speaker driver by using a weaker magnet or a heavier gauge wire to wind the coil. 

Once we understand that everything we do to power is exactly analogous to anything a speaker driver manufacturer does while building the magnet/voice coil interface, it is easier to gain intellectual access to the question of why power influences sound quality over and beyond just noise interference. In fact it is no less profound upon the final result of the audio than the design of the driver. But while there are many parameters to overall power quality, the most important are lack of unwanted noise, lack of unwanted vibration and lack of unwanted DC offset. DC offset would be like having a crooked magnet placed in relation to the voice coil. This would create more force for one polarity than the other. Remember that it makes no difference whether you are cranking a turbine to create electricity or driving a motor with electricity to create movement. In both cases the aspects to quality are influenced by the same parameters, although the methods used to achieve them can be different.