Why do we lean the mixture when the air become less dense? In a carbureted engine at higher altitudes, less dense air means there is less air entering the engine. Wouldn't we want the mixture to be full rich to allow the maximum air into the engine?

I understand that the engine will become rough if we don't lean the engine. I'm asking what is going on? It seems like a decrease in air means we would need to open the mixture not close it.
The decrease in air means less oxygen. If the fuel is burnt at "rich" there would not be enough oxygen in the air to burn some of the fuel, thus reducing efficiency, increasing the risk of carbon monoxide, and adding grime to the exhaust system. That is why "lean" is used for high altitudes, to burn efficiently and safely.