I will certainly try for you Spinner
The underpowered amp can damage speakers by "clipping".
When an amplifier is pushed to create a signal with more power than its power supply can produce, it will amplify the signal only up to its maximum capacity, at which point the signal can be amplified no further. As the signal simply "cuts" or "clips" at the maximum capacity of the amplifier, the signal is said to be "clipping". The extra signal which is beyond the capability of the amplifier is simply cut off, resulting in a sine wave becoming a distorted square wave type waveform.
Underpowering a speaker is likely to damage the voice coil due to the excess heat created by distortion. This distortion, called clipping, is created when the amp is not able to supply the power demand when the volume is turned up.
The signal going to the speakers from the amp is AC (Alternating Current).
(the voltage is alternating too, but they have to call it something ).
This means that to play say a 100Hz tone, the amp will alternate the output voltage (and hence current) between say +12V and -12V (or +1.5A and -1.5A) 100 times a second (the actual voltage/current level depends on the volume you select and the impedance of the speaker, which I've assumed to be 8ohms in the above example).
Now, a powerful amp may be able to deliver a maximum of +/-100V across a speaker, or +/-12.5A into it, where a low powered amp can only deliver +/-40V or +/-5A.
What happens if the low powered amp tries to deliver +/-80V or +/-10A?
The tips of the voltage or current wave are simply "cut-off" or "clipped" - for the duration of the "clip", the voltage/current is steady, which means it's DC (Direct Current - ie not alternating). This is very bad for speakers, especially tweeters (you may think the series capacitor in the crossover would protect the tweeter from DC - it won't in this case, as essentially what you have is pulsed DC, which a capacitor will pass).
Imagine a sine wave, and simply slice off the tips - you'll have what's beginning to look like a square wave - the flat bits now at the tops and bottoms of the wave are the DC component.
DC can burn the voice coil out (if long enough in duration) - and can cause it (and hence the cone) to move too far, which can cause mechanical damage to the speakers motor assembly.
As an amp is turned up closer and closer to its maximum level, the level of distortion increases. So the idea is that if you are using a underpowered amp and have to turn it up near its max to achieve the level of loudness you desire, it is working real hard to reproduce the waveform accurately and may clip. That is why it is better to have an amp with greater power than you will actually need.
A little guide on Impedance (for those who may not understand)
An amplifier is not expecting to "see" a certain load and they are not built to drive a certain load. They have power values stated at certain standard impedances and have a minimum impedance which they are capable of driving, which is related to the amps ability to dissipate heat. An amplifier will generate more heat the lower the impedance attached to it. A higher impedance will make it run cooler not hotter.
Image pushing a big rock along the ground. The size of the rock relates to impedance and the effort required to move it relates to the amps power. When you have a big, heavy rock (8 ohm for example) there is more resistance to movement and you have to push hard and it moves slowly (low power output). If you halve the size of the rock (4 ohm) you don't have to push so hard and you can push it faster (higher power output). Halve it again (2 ohms) and you can run with it! (max power output). The faster you are moving, the more heat you produce. If you get to the point where you are running so fast you cannot keep cool, you will overheat and die.
This isn't supposed to sound patronising, its just an easy way to simplify certain principles.