According to Torah law a person becomes impure only through contact with a "father of uncleanness" and not through contact with something that only has first degree or lesser uncleanness. However, the rabbis decreed that if a person touches something with first degree uncleanness his hands become defiled with second degree uncleanness. If he then goes and touches terumah he disqualifies it such that a kohen could not eat it. At a later period in history, the rabbis added to this and stated that even if one is not sure if he touched something that was impure, his hands are impure. People tend to touch many things (in Hebrew this is phrased—hands are busy) and it would be virtually impossible to prevent one's hands from ever touching something that was impure. To remedy this problem the rabbis decreed that one should always wash one's hands before eating any food. This is the source of the custom that remains to this day to ritually wash one's hands before eating. Today we only do this before eating bread, but originally they would have washed their hands ritually before eating any food.
The issue of washing hands was quite contentious in the ancient world and there are even some passages in the New Testament in which the Pharisees object to the fact that Jesus's student don't wash their hands before eating (see Mark 7 and Matthew 15). Furthermore, people were often meticulous about washing their hands without any ritual connection—it was done purely out of cleanliness. Thus there are many theories as to how the rabbis came up with the idea of ritual hand-washing and the possibility that hands could be purified or defiled as if they were not a part of the body. It is a fascinating topic, one which I partially addressed in my book, The Schechter Haggadah. In my commentary on this tractate we will not explore the origins of ritual hand-washing. Rather we will focus on how these rules manifest themselves in the Mishnah itself.