Well, I have no particular expertise in optics. So I guess I can offer a suitably entry-level explanation...
Anamorphic lenses are ground to a curvature that "squeezes" the light rays passing through them to create an image that's magnified more along one axis than it is along the other axis. Imagine your standard 1x1.33 NTSC "square" image field. Because an anamorphic lens horizontally compresses the light entering it, the final image will contain a 1x1.86, or wider field-of-regard optically squeezed to fit your 1x1.33 image sensor (or film gate, or whatever...)
Ever watch some old movie on tv, where the credits, and maybe certain bits of the film, looked wierdly tall and skinny and spindly and stretched? That's what the squeezed image looks like, when you record it. A hoary overused technique, especially in the 1960s, was to rotate the anamorphic adapter while shooting through it, for this awful motion-sickness inducing spinning stretching distortion in the image...
When the image is projected (on film) the projector is fitted with an anamorphic un-squeezing lens, which takes the light through the horizontally-compressed image on the film strip and optically spreads it to re-create the original field-of-regard on the screen.
When the anamorphic footage is on video, you use a software tool that digitally un-compresses the horizontally squeezed image to restore the original field-of-regard in the center of your screen, with no data (black) at the top and bottom.
That's my crack at it. There are plenty of people posting here with more technical savvy than me. I'm sure they'll correct and elaborate as required...
I just watched a Music Video a co-worker shot and it had some some shots where he was playing around with the Anamorphic lens while recording which gave it that look you were talking about. The cool thing is that they didn't over use the effect which to me looked pretty good.