On the day of Yom Kippur, when fasting, many things are prohibited, including that you can not wear leather shoes or wash hands. What is the reason for the prohibition of leather shoes and washing hands?

The Torah commands us on Yom Kippur to "afflict yourselves". Primarily this means not to eat or drink but our Rabbis explained that it includes other afflictions such as the ones you mentioned. The main reason is to remove ourselves as much as we can from our bodily pleasures and material needs and to make ourselves as holy as we can be on this special day to the extent of emulating the angels. The prohibitions themselves are also an atonement to the sins which were stemmed from our material and physical desires. Shanna Tova