In a nutshell, it allows for up to 128 individual sounds tracks across 64 speaker feeds. This allows directors and theater owners to provide custom sound mixes for individual rooms and localize sounds within the room.
Think of it this way:
Mono - 1 speaker feed - sound come from the source of the video
Stereo - 2 speaker feeds - sounds can come from left, right or both, allowing for sounds to move across the screen.
5.1/7.1 are similar, but the allow for left/right and front/center/middle, etc.
Atmos takes that to a new level with overhead speakers so sounds are actually moving throughout the audience.
The Dolby video in the OP link is very informative if you want more info.