What usually happens to a person's debt at death is that all creditors are notified -- and the outstanding debts are ranked in a hierarchy set out in state law
If there isn't enough property to satisfy the creditors at the tail end of the list, they are usually simply out of luck and out of pocket. Highest on the list are the predictably peskiest creditors, such as those demanding payment for expenses of a funeral and last illness, lawyers' fees, and the IRS.
Credit card companies are generally lower down on the ladder. And you, as the next of kin, will generally not be held legally responsible for such credit card debt unless you agreed to be, for example, by co-signing a purchase agreement.
and paid off from remaining estate property.