I heard professional chessplayers must train physically as well as mentally. Why is that? I heard that both Fischer and Karpov felt that physical training was extremely important to their success and that the Russian coaches also felt a certain amount of physical training was necessary.  Why is that?  How is it supposed to help?  What sort of exercise is supposed to help.
You've heard it right. Physical training is as important as mental training in chess. Thinking is an energy intensive activity and often has a direct effect on the physical body. 
Quoting this scientific american article by By Ferris Jabr on July 18, 2012, brain weighs about 1.4 kilograms, only 2 percent of total body weight, it demands 20 percent of our resting metabolic rate (RMR)—the total amount of energy our bodies expend in one very lazy day of no activity. 

Karpov had lost 10kg (22lb) over the course of 1984-85 Kasparov-Karpov world championship match. Quoting Kasparov from this interview, I pay a lot of attention to my physical condition; to be fit under the terrible pressure of big competitions. Obviously this is becoming more important with age. 

From a study by Speakman JR et. al , being physically fit improves resting metabolic rate (RMR) which is a major contributor to daily energy reserve and thus any increases in RMR in response to exercise interventions are potentially of great importance.

Chess, as you know is a brain intensive activity and thus RMR plays an important important role. Therefore from the 1st and the 3rd paragraphs, it is clear that physical fitness improves mental fitness and therefore beneficial to chess.