You have to be careful to maintain mixture homogeneity when you are creating an emulsion. You are trying to create a perfectly confused mixture of polar and non-polar constituents that will form an emulsion with each other.
The point of mixing the sugar is to increase the surface area of the butter but also to disperse the sugar throughout the butter homogenously. The sugar scrapes against the solid fat and creates tiny air pockets dispersed throughout the butter.
Adding the liquid (egg) slowly gives time for the sugar to gradually be dissolved and allows time for the aqueous solution to be slowly dispersed through the butter. If you rush the eggs, the sugar will be too rapidly dissolved into the liquid. When the sugar starts dissolving in the egg (aqueous, polar), it will create liquid pores in the butter(hydrophobic). As you mechanically mix the ingredients, the liquid pores will come together and become larger. Eventually all the liquid pores come together and completely and undesirably separate from the butter consistuent. If you however add the egg at the right rate, you can still confuse the constituents of the mixture into becoming an emulsion because the rate of the sugar dissolution will be slower and thus the homogeneity of the mixture will be preserved.
The reply about the temperatures of the constituents you have received is also correct for much the same reason. The butter will pull in on itself when you try to incorporate cold eggs which will prevent dispersion and the desired confused polarity cohesion.
I think @boulangere's answer is great. I have only this to add: if your eggs are Large or XL, leave them in that bowl of warm water for longer. If after 10 minutes your water feels cold, replace with more warm water. If I know I'm going to make a cake at 2pm, I pull out my eggs and butter by 8am, depending on the ambient temperature of my kitchen. Cold butter meeting cold eggs will "break" a batter, whether eggs are added all at once or one at a time. Butter and eggs want to form an emulsion, and from that, gain air, so that cake will rise. But eggs and butter and like water and oil. They will not dance together at the party unless they are both the same temperature. Happy cake making!
I'm guessing that you're creaming room temp butter and sugar, then adding the eggs. If your eggs are straight out of the refrigerator, soak them for 5 minutes in warm (not hot!) water, then crack them into something from which you can pour them easily, like a liquid measuring cup. Once the butter and sugar are fully creamed and pale in color, with the mixer running on low speed, tip the eggs in one at a time, allowing each to be fully (fully!) incorporated before adding the next. Adding them gently warmed and one at a time, allows the fats in the yolks to emulsify with the butter, bringing along with them the water in the whites which also bond with the sugar. If the eggs are too cold when you add them, the effectively reduce the temperature of the butter, making emulsion more difficult. Adding the eggs all at once also inhibits emulsion, and results in that scrambled egg appearance.
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The point of mixing the sugar is to increase the surface area of the butter but also to disperse the sugar throughout the butter homogenously. The sugar scrapes against the solid fat and creates tiny air pockets dispersed throughout the butter.
Adding the liquid (egg) slowly gives time for the sugar to gradually be dissolved and allows time for the aqueous solution to be slowly dispersed through the butter. If you rush the eggs, the sugar will be too rapidly dissolved into the liquid. When the sugar starts dissolving in the egg (aqueous, polar), it will create liquid pores in the butter(hydrophobic). As you mechanically mix the ingredients, the liquid pores will come together and become larger. Eventually all the liquid pores come together and completely and undesirably separate from the butter consistuent. If you however add the egg at the right rate, you can still confuse the constituents of the mixture into becoming an emulsion because the rate of the sugar dissolution will be slower and thus the homogeneity of the mixture will be preserved.
The reply about the temperatures of the constituents you have received is also correct for much the same reason. The butter will pull in on itself when you try to incorporate cold eggs which will prevent dispersion and the desired confused polarity cohesion.
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