zibadawa timmy's answer is basically correct, that is the theory. To add some quotes from someone detailing it (in NYT):
The theory builds on Section 4 of the 14th Amendment to argue that Congress, without realizing it, set itself on a path that would violate the Constitution when, in 1917, it capped the size of the federal debt. Over the years, Congress has raised the debt ceiling scores of times, most recently two years ago, when it set the cap at $31.4 trillion. [...]
Section 4 of the 14th Amendment says the “validity” of the public debt “shall not be questioned” — ever. Proponents of the unconstitutionality argument say that when Congress enacted the debt limit, effectively forcing the United States to stop borrowing to honor its debts when that limit was reached, it built a violation of that constitutional command into our fiscal structure, and that as a result, that limit and all that followed are invalid.
OTOH that piece rejects as such
I’ve never agreed with that argument. It raises thorny questions about the appropriate way to interpret the text: Does Section 4, read properly, prohibit anything beyond putting the federal government into default? If so, which actions does it forbid?
There's also the "lite" version that say one has to choose the "least unconstitutional/illegal" measure:
Ignoring one law in order to uphold every other has compelling historical precedent. It’s precisely what Abraham Lincoln did when he briefly overrode the habeas corpus law in 1861 to save the Union, later saying to Congress, “Are all the laws, but one, to go unexecuted, and the government itself go to pieces, lest that one be violated?”
That has been articulated in a bit more detail.
But there is also one paper that purports to detail [more] precisely which actions can be deemed forbidden, by introducing a "substantial doubt" test:
This Note argues that the Public Debt Clause is violated when government actions create substantial doubt about the validity of the public debt, a standard that encompasses government actions that fall short of defaulting on or directly repudiating the public debt. The Note proposes a test to determine when substantial doubt is created. This substantial doubt test analyzes the political and economic environment at the time of the government’s actions and the subjective apprehension exhibited by debt holders. Applying this test, this Note concludes that Congress’s actions during the 1995–96 and 2011 debt-limit debates violated the Public Debt Clause, though Congress’s conduct during the debate over the debt limit in 2002 did not. And under a departmentalist understanding of executive power, a conclusion of this nature would be the basis for the president to ignore the debt limit when congressional actions create unconstitutional doubt about the validity of the public debt.
Interestingly, that paper quotes such interpretations before there was even a [single] debt ceiling as such (introduced in 1917):
In his 1901 Constitutional History of the
United States, Professor Francis Newton Thorpe notes the breadth
with which the Public Debt Clause was interpreted during the
ratification process:
The national debt . . . was held chiefly at the North, and its repudiation, or diminution in value, or any distrust of its obligation, would affect most disastrously the lives and fortunes of the Northern people and would injure our national credit abroad. Its validity was
essential to our prosperity, however great the burden of payment
might prove to be.
Professor Thorpe reports that “validity”—the aspect of the debt
that “shall not be questioned”—was equated with “diminution of
value” or “any distrust” of the government’s obligations.
Also intersting, in Perry v. United States (1935), SCOUTS [of then] hinted it might have perceived debts other than Tresuries as included in that clause, and broadly speaking concerns regarding its integrity:
“Nor can we perceive any reason for not considering the expression ‘the validity of the public debt’ as embracing whatever concerns the integrity of the public obligations.”
(Perry however lost the case in a 5-4 decision. He was arguing that the government debasing the gold dollar was also a violation of the 14th Amendment, but the Court narrowly held otherwise in that specific regard, because the debt was denominated in dollars, essentially.)
On might argue that there is a little historical basis for introducing a test like that, because e.g. events in 1979 did lead to a short-term technical default, in the view of some, despite the official denials (by calling it a "logjam").
Then, as now, Congress had been playing a game of chicken with the debt limit, raising it to $830 billion [...] only after Treasury Secretary W. Michael Blumenthal warned that the country was hours away from the first default in its history.
That last-minute approval, combined with a flood of investor demand for Treasury bills and a series of technical glitches in processing the backlog of paperwork, resulted in thousands of late payments to holders of Treasury bills that were maturing that April and May.
“You hear lot of people say, ‘The government never defaulted.’ The truth is, yeah, they did . . . It might have been small, it might have been inadvertent, but it happened,” said Terry Zivney, a finance professor at Ball State University who co-authored a paper on the episode entitled “The Day the United States Defaulted on Treasury Bills.”
All things considered, the incident amounted to a minor blip. The Treasury had missed payments on about $120 million worth of bills, a tiny amount even then, given the global investment in U.S. debt. Investors, some of whom joined a class-action suit against the government to recover damages, eventually were paid in full with back interest. T-bills, as they are known, continued to be considered a safe investment. Treasury officials both then and now argued that the event was not even a default, but merely a delay caused by the internal logjam.
