I'm not 100% sure, but I think that's what following does.
Subscribe to players and get notifications when they play
I’m under the impression that following players only gives you notifications when they post in the forums, write blogs and articles etc. The staff mentions it here:
https://support.chess.com/customer/en/portal/articles/1444797-how-do-i-follow-or-unfollow-another-user-
It would be awesome if i could «subscribe» to player accounts and get an notification from the app when those players are playing on the site. So if i have subscribed to Magnus Carlsen, and he randomly joins a Titled Tuesday for example, the app would scream «Hey dude, Magnus is playing right now».