What is the rationale behind the prohibition of not wearing leather shoes on Yom Kippur?
The shoe symbolizes the physical body. Just as the shoe encases the lowest part of the body and allows it to ambulate in the world, so too the body encases the lowest level of the soul and allows it to ambulate and relate to the physical world.
Therefore, whenever G-d wants a person to relate on a totally spiritual level, ignoring the body, He commands him to remove his shoes. This was true when G-d spoke to Moses and to Joshua; it was true for the kohanim in the Temple in Jerusalem, and it is true for every Jew on Yom Kippur. We ignore the physical for one day a year, and to symbolize this we remove our leather shoes. Leather specifically, because it came from a living creature and hence symbolizes the body in a much more graphic way than other materials.
The shoe is also removed in a ceremony called "chalitzah," as follows: If one of two brothers dies childless, it is a mitzvah for the widow and the surviving brother to marry each other. If the brother refuses, then the widow is to remove his shoe, signifying that he does not deserve physical comfort or even a body, because he refuses to give a physical form to his deceased brother's soul.