When adding eggs to scratch cake batter, why are you supposed to add them one at a time?
The key thing here is a substance called Lecithin and it is found in your Incredible, Edible Egg!
Doing the adding of the egg one-by-one, plus a few other things are...

in the service of creating a stable emulsion of liquid and fat in the batter. When this is done correctly, the cake will have a springy, even crumb, great flavor, and light texture. If the batter is not properly emulsified, the resulting cake can be uneven and flat, flavorless, and have a heavy texture. It's even possible for the cake to &quot;fall&quot; or sink into itself when baking.
An emulsion is the suspension of small globules of one liquid in a second liquid with which the first will not mix, such as oil and vinegar. To create an emulsion, we slowly add eggs to the butter mixture, one at a time, beating rapidly to suspend the water from the butter and egg whites in the fats from the butter and egg yolks. Lecithin, an emulsifier found in eggs, helps to stabilize the emulsion, as does aeration from steady, rapid beating.
Adding the eggs slowly makes it easier disperse the liquid, and promote even suspension of liquid in fat.

So to recap, liquid and fat do not mix together. The egg has Lecithin that acts like a bond that keeps those 2 together. By flooding the mixture with the substance that will act like a binder (egg), you actually make it harder for all the water in the butter / eggs to bind to the fat in the butter / eggs.
Picture of Emulsion


Take note that in this picture, the oil to make the mayonnaise was added too fast in the failed picture. Take a look at thesproatingseed.com link below for even more information on emulsion!
This answer was made possible by SeriousEats.com, IOANA.BLOG, and TheSproutingSeed.com