The federal so-called "debt" consists almost entirely of unexpired treasury securities. The debt limit is an artificial constraint on the allowed dollar amount of unexpired treasury securities. Once that limit is reached, the Treasury is not permitted to sell more T-bills and such. So?
Well, recording treasury securities sales amounts is an artificial mechanism for crediting the balance of the Treasury General Account, the "checking account" from which the Treasury supposedly spends. Another artificial mechanism for crediting the TGA is to record the amount of tax collections there. The Treasury sells T-bills (securities) equal to the amount of government spending over and above tax collections in order to to not overdraw the TGA. So?
If outstanding securities hit the debt limit and the Treasury is no longer allowed to sell securities, then the TGA cannot be credited enough to cover federal expenditures. So?
So technically, the Treasury can't spend.
The trouble is -- all these constraints, the TGA itself, the rule that the TGA cannot be overdrawn, the crediting of the TGA with tax dollars (which actually cease to exist once collected), the crediting of the TGA with T-securities sales, and the debt limit itself -- these are all artificial, self-imposed constraints that have no practical purpose except to discourage congress from spending frivolously. They are carryovers from the long ago abandoned gold standard. The federal government does not need to tax and borrow in order to spend.
There is no real constraint on how much the federal government can spend because it does not spend existing "money". It, through its agent the Federal Reserve, spends by creating bank reserves and deposits in commercial banks and there is no real limit on how many reserves and deposits it can create. Remember, the TGA is an artificial, self-imposed constraint, so any shutdown of federal spending related to shortages in the Treasury General Account, especially one caused by reaching the debt limit, is totally optional.