So what does the milk do? The only difference I can see is color - without the milk the eggs are a brighter yellow.
What am I missing?
Well, it makes the eggs go further for one...
But it also produces softer, creamier results. You're moving toward something like a custard or quiche. If you like your eggs very stiff, this is probably a bad idea.
If you've never made scrambled eggs the Gordon Ramsey way you're really missing out. He calls for fresh cream, but I use sour cream and find it works just as well and gives a nice tangy flavor. Great video where he demonstrates the technique: http://videosift.com/video/Gordon-Ramsay-s-Perfect-Scrambled-Eggs
According to him, one of the reasons to add milk or cream is to cool the eggs down so they don't keep cooking after you remove them from the heat.
It's worth learning to cook excellent scrambled eggs without the milk and cream, in my opinion.
Traditionally, (well, at say Cordon Bleu in the 1950s), cream would be added to stop the eggs from overcooking once they were properly done. And, like people mentioned, they get creamier as well, but the cream would be cold and added at the end; its primary purpose was stopping the overcooking.
If you heat slowly, shake gently, and treat them kindly, scrambled eggs can be totally freaking fantastic without any additives. Start there.
According to Cook's Illustrated, the fat in milk or cream will actually separate the protein strands from the eggs, resulting in fluffier eggs. And fats give a smooth taste to food that you can feel on your tongue.
Scrambled eggs without dairy fat in them are very, very easy to overcook.
Using milk in scrambled eggs results in eggs that are moist and, er, creamy. Texture-wise, they come out softer (some might say "gloppier") than eggs without. Flavor-wise, they're a bit more mellow and richer. The downside is that they don't come out as fluffy (unless you're just using a small amount).
I've known people who think milk in scrambled eggs is the devil's additive. I really like the softer texture and the difference in flavor, though. Try them yourself, and you may! Try them and you may, I say!
There are two philosophies on cooking scrambled eggs: some prefer them cooked slowly over low heat, while others swear by a very hot pan. If you cook them slowly, milk or cream is primarily there to make them tender, and perhaps to prevent overcooking.
However, none of the other answers have mentioned fast cooking. If you pour raw scrambled eggs into a very hot pan, they will begin to cook almost instantly. In that case, any added liquid (even water) will add to the boiling effect near the pan surface, producing steam that will separate the protein bits with air pockets and fluff the eggs.
I have never noticed a significant effect on fluffiness by adding liquid in slow-cooked eggs. But for the fluffiest scrambled eggs possible, cook on high heat with a bit of liquid added. Just be very careful to keep the eggs moving and remove immediately while they are still slightly undercooked, or they will dry out. (The eggs will continue to cook even out of the pan.) On the other hand, make sure they are cooked enough, or they will "weep" liquid. It takes a little practice -- with a very hot pan, even 5 seconds can make a significant difference, so have your plate ready.
The fast scrambled eggs technique is more difficult, but it's time-efficient, and the extra liquid added can produce very fluffy eggs. (For the record, the same technique can be used for extra-fluffy omelets -- very hot pot, a little liquid, keep things moving and remove promptly.)
My guess it makes it softer and more moist. I usually don't put milk in it but a good piece of butter and sour cream which gives it a great creamy texture.
The addition of milk is to make it fluffier and lighter. In my opinion (emphasis on "my"), it is like cooking with training wheels. I have never liked the watered down flavor of eggs done this way and I much prefer the denser flavor of eggs sans milk. You do have to be more attentive and make sure the eggs are not overcooked. It's a bit tricky and you have to remove them just before they look quite ready, and they will become perfect by the time they cool a bit.
Never add milk or cream during cooking, it will separate the mix and make it watery. Only add cream or creme fraiche after off the heat.
Never add milk or cream to eggs because it takes away from the flavor of the eggs also it doesn't make it fluffier or anything so basically it's a waste.
I add about a quarter cup of 3.25% ( homo ) milk to 3 XL eggs; put them in a hot pan, on medium, with some melted butter, about 1 tsp. in it. Soon as they start to bubble, I begin stirring "slowly", with a spatula. Once they are about 3/4 done, I put a glass lid on them, so I can watch them because I don't want all the liquid to evaporate out of them. Then, after they have firmed up, I remove the lid, break them all up and serve them hot cause I like my food very hot. They still have some tasty juice left that I can dip my toast in. The milk keeps them soft and fluffy, even though I put my salt, pepper,hot sauce and green onion ( or chives ) in during the mix. I find that putting the salt on after they are cooked, gives them a tart metallic taste; but for those that may prefer, the spices can be added after. Well that's my 2c worth.
I've always used milk in my scrambled eggs (mixing the milk in with the eggs before cooking). The soft and fluffy results speak for themselves.
It's definitely an improvement over adding water. For years, my grandfather made breakfast at his house. He didn't use milk or cream in his scrambled eggs--he used water--and never strayed from that. To say they were unappetizing is an understatement.