Beating egg whites . Directions are often room temperature eggs and very cold bowl bowl and beaters. Why and does it make difference?

Yes, in general, room temperature eggs will whip up better than cold eggs. But cold eggs separate more easily. So when I'm whipping egg whites, I'll either use eggs from the bowl on the counter, or separate eggs from the fridge as soon as I take them out and then let them sit out at room temperature.
As for the bowl and utensils being cold, that holds true for whipping cream as well. In addition, it's important that there's no grease on your bowl or whisk because that can hamper your efforts to whip your egg whites.
Lisa Steele
Author of The Fresh Eggs Daily Cookbook

I've actually never seen that. I've read that eggs are easier to separate when they are cold. I've also read that it makes no difference at all whether the egg whites are cold or at room temperature when beating them.

Here is an article on Food52 about beating egg whites that you may find helpful:
https://food52.com/blog/17380-before-beating-egg-whites-read-this

I hope this helps. ;o)