Yes, in theory, it's supposed to combat this:
Practically, it's about as well thought out as Leeroy Jenkins' ill-fated dive into the Rookery of Blackrock Spire. On the surface, it's the stupidest thing you've ever seen, but also like the Leeroy Jenkins video, there may be more going on that there appears to be (the video is staged, for comedy).
I'm inclined to say that although I don't buy any of the conspiracy theories regarding this change, I also don't think that this is to fight forum trolling so much as it is to tie all of their products together more tightly, creating easier transitions from one to another in the days to come, when Blizzard will inevitably release a new MMO.
Thankfully, this change will not apply retroactively to old posts. I guess I'm not going to post in the forums any more, though (not that I really ever do).
Even more disturbingly, some addons can apparently read (and repeat to other people) your real name thanks to in-game RealID. Awesome. Guess I'm killing all my addons for the moment (not that I use so many).
But what's really interesting about this is that it's the biggest forum furor (which are always hilarious in and of themselves) I've seen since the second coming of Bus Shock, and impressively, this clusterf*ck may be even more... shall I say... Cataclysmic?
P.S.: I predict that by week's end, either Blizzard has addressed this issue or it has become clear that people really don't care that much. Still, it's an impressive storm brewing on the forums.
Update #1 (7/7/2010):
Well, the sh*t is really flying now, and Godwin's law has hit the forum hard regarding Blizzard's liberal application of the banhammer. Inevitable V for Vendetta comparisons are being made, as they always are whenever the internet gets angry. The official thread on Battle.net changes is over 1,000 pages long. Blizzard has also made clear that this change is part of their long-term plans, and seems to be holding firm on the issue. It'll be interesting to see where things go from here.
Some interesting points have been raised in the debate, the most valid of which, I think, is that this change has some disturbing ramifications for those whose real names reveal a great deal about their identities. Of course, the standard response from supporters has been "don't post, don't have your name revealed," but while technically true, this answer is laden with unfortunate implications about how people who fear being victimized should be silent.
Update #2: And lo, by Friday night, it has been retracted. Real names will not be displayed on the forums.