That is true when you close an account (like paying off the last $10k on a home loan), but I don't think that is correct when paying a credit card down to a $0 balance and leaving it open.

The reason is that your credit score is impacted by both your available credit (higher is better) and credit utilization (lower is better). When you pay of the last of a home loan and close that account, your available credit goes down and your credit utilization goes up (assuming you had any other debt). Both of those hurt you. When you pay the credit card down to $0 and leave it open, your available crediot stays the same and your utilization goes down.