Many thanks to Dustin for sending us the screenshot of the process (see a larger version). You can see that the NPC gives you a little background about the feature, and then the confirmation box comes up asking if you're sure that you want to spend all that gold.
Obviously, this is the point where you click "Accept" and dance around for joy that you now have two specializations.
We'll have more about the exact mechanics and workings of dual specializations as the day goes on. In the mean time, check out the WoW Insider Guide to Patch 3.1 and the PTR Patch 3.1 Patch Notes for more information.
Patch 3.1 brings us Ulduar, dual specs, significant changes to all the classes, and more! We've got you covered from top to bottom with our Guide to Patch 3.1.