Think of an amplifier stage as a set of three lines:
1. At the top, the maximum positive excursion
2. The middle line is silence
3. The bottom line is the maximum negative excursion
if you try and go higher than line 1 or lower than line 3, in the analog world there just isnt enough voltage in the system to handle it any further (for example) and in digital, there is just no information left to represent that value
Heres a snare drum hit, operating in the legal range
Note the clean, detailed transients.
Now here is the same snare hit, with the volume raised to exceed the range of a 16 bit plugin placed in the path
Those squared off waves, just like the squared off crap in a modern pop40 CD, is where that nasty distortion comes from