What exactly is textural density??

It’s perhaps described as a combination of textural density tone and weight IMHO. A note has a beginning middle and end and it is the ability to fulfill these

@erik_squires 
Anything in music that doesn’t sound like an AM radio. And then like anything else there are different degrees. 
Hope that helps!

John Atkinson's writing may be clearer, but his conclusions are inscrutable.

+1 to cheeg!
We've all used adjectives like "threadbare", "thin", or the old standby, "tinny" to describe sound reproduction that reduces the "richness", "color" or "warmth" of the music.  This is just another positive adjectival phrase in the same vein.  All this for those of us with BA degrees not EEs. 

“Textural density” in audio is the (partly successful) appropriation of a term in music which refers to the way that instruments/voices and their respective musical lines in a musical composition are used. Each musical line is a “layer” and the “density” of the composition refers to the overall character of the sound as determined by the number of layers employed in the composition. One or few layers results in a “thin” musical texture. Many layers, a “thick texture”.

I think one can then extrapolate and in audio, if one thinks of the naturally occurring harmonics in musical sounds as “layers” themselves, a reproduced musical sound which expresses an accurate (natural) number and combination of harmonics can be said to be “texturally dense”. One that is lacking in full harmonic expression can be said to be “thin” or “threadbare”.