I'm sure there are a lot of people who might think it means it was rated Class A by Stereophile!
But I think the best way to think of it is as was stated in the link above, the circuit is biased so that the signal always stays above "zero", or as they put it, more positive/less positive, as opposed to positive/negative. It does usually put devices into their most linear region, and eliminates dreaded crossover distortion.
Funny, I remember my test from tech school years ago where we had to remember Class A, Class B, Class AB, and Class C amplification with transistors. Not that I have used it much since then, until this little headphone thing came along!