In general I think one chooses to be invisible when for a reason or another there isn't the possibility to reply to others contacting this person. For example, if you log in only to check a thread but have no time to reply to messages or start a conversation with some other member in private, you can choose to be invisible.
I've also seen people who are under invisible mode because they are "cowards": I'm referring to those people leaving the forums with harsh words or that before leaving offended somebody, then they come back but decide to be invisible to make others think they really left as they said. They clearly don't want to appear like the ones who are coming back after all... But they don't think about the fact admins can see them