Why is water often added to the eggs when making omelettes?

Eggs are already 3/4 water anyway!

By mixing in a small quantity of extra water before you cook the eggs, you are slowing down the cooking process by making more water available that has be evaporated. This keeps the cooking temperature to less than 100°C (212°F) for longer, therefore increasing the the time for the egg proteins to foam and expand before setting

The amount of water you need to add depends on; personal preference, the type of egg, and how old it is. Older eggs generally require a little more water

Adding skim milk will enhance this process slightly too. Adding extra fat will generally not enhance this process

BONUS TIP To make even more spectacular omelettes place a loose fitting lid over the pan to increase the steam exposure all around, and let the egg fully develop