Why is it possible for bare feet to touch red-hot coals without getting burned? The coals start out as pieces of wood. The wood consists of lots of carbon, some "volatile organic molecules," and water.
A volatile organic molecule is a carbon-based molecule that evaporates when you heat it. Gasoline is a volatile organic chemical. We see these volatile organic molecules from wood as smoke rising from the fire. The heat of the fire evaporates all the volatile organics, as well as all the water. Because they've been burning for a while before the stunt, the coals have burned down to nearly pure carbon.
If you were to pick up one of these pure-carbon coals, you would notice that it is extremely light. Carbon is a lightweight element -- that's why carbon-fiber bike frames and tennis rackets don't weigh very much. A coal is mostly lightweight carbon atoms and air spaces (it does contain a few other elements, like potassium and calcium -- that's what left behind in the ash).
This lightweight carbon structure is a poor conductor of heat. It takes a relatively long time for heat to transfer from the glowing coal to your skin. If the coal were made of red-hot metal, heat transfer through conduction would be almost instantaneous -- you would get a severe burn.
Now, add to that the fact that ash is a very good insulator. People used to use ash to insulate ice boxes and such. The red-hot coals covered with ash transfer their heat even more slowly because the ash acts as a layer of insulation.
Then there is the short time span. Heat transfer from a red-hot coal is slow, but it still happens. If you were to stand still on the coals for several seconds, you would definitely get a burn. By walking briskly, you limit your contact with individual coals to a very short time span. You also get across the bed of coals very quickly, and that limits your total amount of coal time. So, your foot never gets hot enough to burn.
That's the whole story! For more information on firewalking and related topics, check out the links below.