Can someone explain what damage per gold is?

You do 2000 damage in a game u got 2000 gold. It is 1 damage per gold

Just a way to measure the use you gave to the resources (The higher, the better)

Damage done to enemy champs divided by total gold earned

The main take away you should have for it is if you have high gold every game but much lower damage per gold than the average, then you may either be spending too much time PvE farming or you die at the times it is important to use your gold advantage.

Of course there could be other reasons. You would have to review your games to be sure.