When an amp does begin to push the limits of its normal operating parameters, depending on the amp's design, it can begin to compress the lower frequency signals, it can start distorting, or it can just lose a bit of the naturally dynamic nature of the lower frequency reproduction.
Musical signals are very dynamic in nature and as GRoberts said, the lower frequency notes are very demanding on electronic circuits as far as power demands and handling transient bursts of sound.
Headroom allows the amp to have reserve power to handle the bursty spikes of low frequency information and still remain inside its optimal operating parameters.
The flip-side is if you get carried away with EQ or do something like hit a string against a polepiece on your bass guitar's pickup (been there/done that), your amp is going to send potentially dangerous levels to your cab(s).
Normally, if you listen to your cab, you can tell when it is being pushed too hard and you can adjust the levels to compensate.
I do love headroom, but you have to be careful how you use it.