They don’t. A developed country does not need to keep growing.

A country that made debt needs to keep growing.

Usually countries make huge debt to buy infrastructures, invest in research, or education. This fields are very expensive but are what make you progress. Making debt for this is an investment in the country’s future. The problem is, you have to pay back the debt. Developed or not, if you make a debt and fail to progress, you will not get the extra profits to pay back the debt.

The fact that developed countries are also in debt, is a circumstance. A coincidence. Poor countries also may have a big debt, and same as developed ones, they have to grow to pay back the debt or end up more in debt.