Water vs olive oil when sautéing
In terms of sautéing, the simple answer is that using oil is going to let you develop fond, i.e. the tasty brown stuff, on your veggies whereas cooking only with water will essentially boil/steam your vegetables — and perhaps give them a little char, as well.

In cooking, both oil and water are basically just things you use to transfer heat to food. They are both useful in their own way, depending on what you're cooking.

In practical terms, oils/fats can serve a couple other purposes, too, like:


Help extract fat-soluble flavor compounds from ingredients, like herbs and aromatics.
Keep things from sticking and burning in your pan.
Add flavor.


In a hot pan with only water, your friend was probably boiling/steaming their vegetables — not truly sautéing, in which you really try to avoid steaming. If the water runs low, you risk burning your veggies when they come in direct contact with the hot metal of the pan.

This isn't necessarily a bad thing; it depends on what the food is and how you're trying to prepare it. There are better ways to steam veggies, but hey, whatever floats your boat.

Cooking for Engineers has an illuminating article on the science behind heat transfer. Learning about the Maillard reaction is also a good read. And others more experienced than myself can tell you a lot more, I'm sure. Hopefully this is a sufficient "short" answer.