Why do potatoes discolour when peeled?
Short answer: Potatoes oxidize.

Slightly longer answer: Potatoes contain an enzyme called polyphenol oxidase (the same enzyme responsible for browning in apples), which when exposed to oxygen in the air turns that brown/grey color.  That process is called oxidizing.  Because oxygen is required to, well, oxidize, any means of preventing the air from coming into contact will keep them from discoloring.  Submerging in water is just an easy and convenient way of doing that.  Also, polyphenol oxidase is neutralized during cooking, so cooking the potatoes immediately prevents the discoloration.