Spark Notes

Obama came into office wanting to cut entitlements (Social Security, Medicare/caid).

The Republicans, concerned with getting reelected by frothing, cretinous Teatards have taken a principled stand against raising the debt ceiling, something every single incumbent has voted for in the past.
Not voting for it could mean the collapse of the US economy with all the attendant damage that would do to the world.

Of course there are ways around that; Obama could simply declare it unconstitutional not to vote for it, citing the 14th Amendment, among other solutions, but he has decided not to bother with such routes.
Instead, he dangles massive spending cuts including to the aforementioned entitlement programs while asking for a pittance in tax revenue from the GOP, and of course military cuts are not even considered...
He's being the adult, the better man, he's compromising~

Of course, the elephant in the room is why all those who bankroll the GOP and the Democrats, the various corporate entities and the super-rich who would be harmed (but by no means be the worst affected) by a default on American debt would even allow their lackeys in Congress to actually implement such a threat.
And the answer is: they wouldn't (Boehner's postponed the vote, even). It's all theater, a convenient way for everyone (except the vast majority of the American people) to get what they want; less government handouts to the most vulnerable Americans, less money being paid by the wealthiest Americans, and maintaining a facade of the wise elder statesman, valiantly holding off the barbarians at the gates, while the barbarians get to pretend they're upholding real American values.
Oh, and they get to sack America again.