I know it means the US cannot loan any more money but what does that mean for the American citizens?
Not exactly. It means the US couldn't contact any more debt than it's in already. Problem is, the fed uses debt (selling bonds) to pay off ... well, other debt. The government would be unable to pay bills it has already incurred. That means the government's cost of borrowing would go through the roof, much like a person who missed a car or mortgage payment would see the interest rate on their credit card suddenly jump to 29% or something.
It might also mean that the people who hold U.S. debt (that is, who own bonds, treasury bills & so on) start dumping it because they no longer believe they'll get paid. Essentially the cost the government has to pay to do anything would shoot up. For reasons I only vaguely remember this would maim the value of U.S. currency, making everything more expensive for everybody.
The Treasury Department has enough cash on hand to keep paying the bills until (coincidentally) Hallowe'en. Then they've spent the last of the change they found in the couch cushions & things get really scary.