Pretty much what the title says. Occasionally I'd like to not be connected to my friends list and/or simply appear offline to my friends. Anyone else just want to play alone sometimes? I know I do!
For example, sometimes you have friends who are reaaaally bad but get reaaaally upset if you solo queue for ranked. Of course you don't want to hurt their feelings and say, "No I'm playing by myself" because you get the response "why?!" and then.. well. Awkward.
Perhaps you have an IRL friend who is online 24/7 monitoring their friends list so you can't escape and play with someone else? "Don't play with those ***** PLAY WITH ME *****whinenag" is something nobody likes dealing with.
There are dozens of other reasons for wanting this feature, I'm sure several are coming to your mind as you read this, and hopefully no one disputes the use one could get out of this particular function.