Jonathan Chait explains the Republican strategy, not just with regard to the current shutdown but with everything, lo these past years.
So here we are: Obamacare is open for business and the United States of America is not. Lovely. Playing Armchair President for a minute, I think Obama oughta hit them hard on the point that if the entire House of Representatives were to vote on a clean spending bill, it would probably pass. And the U.S. government has been shut down, a lot of damage has been and will continue to be done, because John Boehner and/or the forces holding him at knifepoint, won't allow that vote of the entire House.
I do have this mental image (with sound) of Boehner on the phone to Obama or maybe Harry Reid, saying, "Look…you've got to give me some sort of concession we can hold up as a victory so we can back off this position."