What they're referring to is blocking of site access on NIPRNet, which is the "unclass" side of US military network operations, but is still subject to additional scrutiny and a strict requirement that no information that has been classified be stored on connected systems. This is standard protocol bordering on the boring for office communications in the military, and is absolute non-news.
Nobody is actively working (well, okay, not openly working) to restrict communications viewed by active duty DoD personnel on their personal computers while utilizing Internet connections not-uplinked-in-the-barracks-or-other-stupid-places-where-you-know-your-traffic-is-being-logged-shipmate. Military personnel are keenly aware that they face serious legal penalties for improperly accessing and or disseminating classified materials. This is not difficult to understand.
It's worth noting that in this particular case, I firmly believe Snowden acted as a patriot and is absolutely not the traitor he's being painted as by the administration and various members of Congress. I say this as a former service member myself (Navy) who also held a TS/SCI clearance. This young man exposed wholesale disregard for our Constitution on a massive scale, and it's been happening at an increasing pace for about twenty years. I ardently hope he finds asylum somewhere safe.