It's not painting, exactly. It's a wax resist technique, like batik. You use a tool most typically called a kistka to write with melted wax all over what you want to stay white, then it goes into a light colored dye bath - say yellow - then after drying you write on anything you want to stay yellow, then the next bath, etc, all the way up to the darkest - red, or black, most often. Then after that dries, you melt the wax - can do it in an oven, or often you hold part of it in the side of a candle flame until the wax melts, then rub the wax off with a cloth or paper towel, until the whole egg is exposed.
http://www.pysanky.info