Great now no one is going to talk to me now.... q_q. But I understand this is an option a lot of people want I think it should be implemented. And actually, I have felt like "going invisible" a few times as well.
There are a lot of people who can't handle being told- "I don't really want to talk to anyone right now." A LOT of them. They take it personally, they get offended, etc. Easier to just avoid the conflict.
Also, while you do have control over who has you as a Real ID or BattleTag friend, you don't have control over people who add you to their normal friends lists. I don't mind helping out new characters, but I like to have my time to do my own thing too. I've definitely had people who added me because I took the time to run them through something, help them with a quest, show them where to get a pet, etc., who then asked me all the time for runs, can I get a guild invite, can I borrow gold, can I get free stuff, etc. Often it's pretty obvious that they're kids, and so I don't want to be mean or hurt their feelings. Being able to look offline would make that easier.
EVERYONE gets pissy over something that someone else thinks is nothing (for example, some people might get pissy about a toggle switch on your friends list that lets you go offline, and I would call that getting pissy over nothing). If you refuse to associate with anyone who doesn't feel the same way you do about whether or not to take it personally when you just refuse to talk to them, you are going to spend a lot of time alone. Also, like I pointed out above, you can be on people's in-game friends lists whether or not you want to be.
About fracking time.
Interesting feature, but I've got some questions:
It`s nice when you just wanna play alone, some ppl botther you even when you say "i`m busy".