As a foreigner the whole government shutdown is a thing of wonder: how does it work exactly? For instance, is there no police anymore? How does Congress stay in session during a shutdown? Where does electricity in federal buildings come from if the Federal govt can't pay for it?
However, the military is entirely part of the federal government, as are many other organizations. In the case of not having a budget, the government is required to shut down all functions that are not essential for safety and security. So things like the military, the TSA, and so on can continue to function, but office workers doing jobs like allocating the funding for highway construction are furloughed, put on temporary unpaid leave from their jobs.
This is, of course, terrible for the efficient functioning of government; when the impasse is over, they will have to come back to their jobs and try to catch up. Oh, and in the time leading up to the shutdown, many of them would have been busy preparing for the shutdown, documenting which functions would be considered essential and thus able to continue during the shutdown; because the deadline was no surprise, and there had been many temporary funding bills leading up to this point, there was always a pretty high chance that this shutdown would happen.
Of course, this inefficiency plays well into the politics of one of the parties, who hold a religious belief that private enterprise is always more efficient than government, and so playing games like this that increase its inefficiency help to sell their case and get government functions sold off to private enterprise, who just so happen, in so many cases, to be some of the largest donors to campaigns of this particular party.
