You’re missing the second part of that.
But neither the United States nor any State shall assume or pay any debt or obligation incurred in aid of insurrection or rebellion against the United States, or any claim for the loss or emancipation of any slave; but all such debts, obligations and claims shall be held illegal and void.
What this meant was that the debt of the united states was valid debt, payable by the united states, but the debt of the confederate states (or , any state or nation that takes up arms against the US) will not be valid US debt.
Basically what was happening was that former confederate soldiers and slaveholders and just general people were billing the US government for costs incurred as a result of the civil war. This was a way of stating those debts were not valid debts and thus could not be actioned on.
