Why would heating left-overs in a ring heat them more evenly?

I believe the unspecified context of this question is cooking in an oven or microwave. No matter what your cooking method, the heat is introduced at the outside of the food:

• Microwaves only penetrate a short distance; the rest of the cooking through to the center is by conduction (or convection, if the food is liquid)
• Conduction/convection in an oven only interact with the surface of the food
• Radiant heat (infrared essentially), one of the main cooking modes in an oven, again only affects the surface

By creating a ring shape, if it is practical, you create more surface for the heating modality to interact with the food.

This is the reason for an angel food cake pan or bundt pan having a tube at the center--it allows more surface area to be involved in the cooking process.

Also, with the metal pan, the metal has a higher conductivity of heat than does air. This is why parts of the cake in contact with the pan (or a non-insulated metal cookie sheet, or whatever) generally brown more than the parts exposed to the air.

However, in practice except for things cooked in tube pans, I wouldn't really worry about this affect. If you can stir occasionally, it becomes close to moot.

Note that in pan frying or sauteing or or griddling or stir frying (depending on the term used in context), the main surface area that is relevant is the bottom of the food, that which is contact with the pan. A ring doesn't help much here, compared to spreading the food out.

I looked at the first part of that thread. It does not seem to be a hot bed of science and knowledge of cookery.

As SAJ14SAJ mentioned, this shape will be preferable to a simple heap when used in an oven. But actually, a spread-thin sheet of food will be better than both heap and ring when you are using it in an oven.

My guess is that this shape is supposed to be used on a gas range with a rather large burner, or on certain types of induction stoves whose induction coil is ring shaped, in combination with a thin pan. In both cases, there will be a ring-shaped area of pan bottom much hotter than the rest of the pan. Putting the food on that area will give you even heating in the bottom of the ring, and will work if the ring is not heaped too high. The higher the ring becomes, the colder the top part of it, and at some point, the food would get hot quicker if put in the middle and on the sides of the pan, in contact with the not-very-hot parts of the bottom, than high above the hot ring area. And of course, you can't really stir when you are using the ring.

Conclusion: it is better for the specific context of ring-shaped heater, thin pan, small amount of food and a busy cook (who can't be bothered to stir). If at least one of these conditions is missing, spreading it along the whole bottom is preferable.