You take out the rings so that the bottom of the pot is heated by the air from the fire.
If you put the pot directly on the metal, you can only transfer heat by conduction, but to get efficient heat conduction the contact area has to be large. However, in the time that these stoves were used, pots, although they generally had a flat bottom, were not precisely engineered but even often hand-made. Also pots were used for a long time and tended to have a lot of scratches and bumps after a while. And the stove itself is cast-iron which is also not perfectly flat.
So the pots would not sit very evenly on a metal surface and the contact area would be small, making heat conduction quite inefficient. Such pots with uneven bottoms are better heated directly by the hot air through convection and radiation from the flame. Modern pots designed for electric stoves have a very smooth bottom. But note also that now induction heaters (which generate a current within the pot and heat it directly) or infrared stoves are becoming more popular, as they don't depend on heat conduction between two (possibly uneven) surfaces which is notoriously difficult to do efficiently.
This is based on own experience with these stoves. When you finished cooking and wanted to keep the food warm but not heat it any further, you'd move the pot onto the metal. Even though the metal was also burning hot, it didn't burn the food.