How does it monitor the activity. Not how it keeps score, did you even read the question?
Activity.

I think it's measured by how many posts you make in the forums on that day. The rankings for the most active member are listed here:
http://www.chess.com/members/search?mostactive=1

See GeorgeBlackChess? Even though you might make more posts on the forums than I do (though I'm catching up quickly), my posts are far more helpful than yours.
How does chess.com monitor activity? How does it, in other words, determine the most active members?