Once the government hits the debt limit, in theory it can no longer borrow and since the US runs an annual fiscal deficit, that means it may soon run out of money and might temporarily default on its obligations, including paying government staff and military personnel.
Failure by the US Treasury to raise additional debt to cover interest payments on its existing debt could result in a default on the government's debt obligations. This would have severe repercussions for the US economy and global financial markets, potentially leading to a credit rating downgrade and even a government shutdown.