I haven't written about this recently, so just a reminder:
At some point after August 2, if the debt limit is not raised, the President of the United States will have no choice but to break some law or obligation. Remember, Barack Obama has no legal authority to stop spending money that's duly authorized appropriated by Congress and signed into law. Nor is he allowed to fail to pay contractors who have valid contracts with the government. Nor is he allowed to not pay interest on debts that have been duly authorized, or redeem bonds that come due. And he won't have enough money to do those things, and will not, per the debt limit, be allowed to borrow in order to do so.
That's the context in which "invoking the 14th" (or it's playful cousin, the coin thing) is generally mentioned. Essentially, the problem is that the law (that is, laws passed by Congress and signed by the president) give contradictory instructions: you must spend this much more than you have available, but you're not allowed to borrow to do it. Given that the president must, in that situation, violate the law (or come up with improbable interpretations of the law), better that he do so in best keeping with the spirit of both Congress's most recent actions and, meanwhile, avoid or minimize economic disaster.
Will Obama choose to blow through the debt limit? I have no idea. There are both legal and political risks if he does so; the question is whether those are more, or less, important than the legal and political risks for not doing so.