To sit for someone else they need to add you as their sitter as described in "How do I add a Sitter?" When they have done this you will be told that you are a sitter for their account on the "Account" tab of your profile. 

You will also be told what rights you have as a sitter for that account, and you may remove yourself as a sitter by pressing the red . Note that you may only sit for two other players at a time.

When you are added as a sitter for another account you may login to it by logging out of your account and then logging in with their user name but your password. You will be granted access as if you had typed their password (so they do not need to and should not share their password with you).