The phrase has very little to do with the work of famous Russian physician Nikolay Sklifosovskiy. It originates from the movie Kidnapping, Caucasian Style. Balbes, an antagonist of the movie played by Yuri Nikulin, exclaimed: "Make it brief, Sklifosovskiy!" to Shurik and his friend, the protagonists, who, having disguised themselves as vaccination nurses, commenced a long lecture about the benefits of immunization.
It literally means "[Make it] briefer, Sklifosovskiy."