Here is my undertanding.

In a piano-bass-drum trio, the bass normally plays the root. If the piano player doubles the root, the sound will be heavy. So for a lighter, more airy, sound, the pianist omits the root.

Well, that's a start. I am not quite sure why it is the 3rd and the 7th that define the chord, or why the 9th is added, or why the 5th can be replaced by a 13th. I would have thought the 5th was pretty important to the sound of the chord. So what about it, Scott?