Sapolsky gives a quick version for the commencement speech. In his book, Why Zebras Don't Get Ulcers (PDF), this is explained in further depth. It's not that the brain burns 6-7000 calories a day, it's the various stress responses in the body that are on the whole greatly on par with top athletes. You need to be in really good shape to play high level chess.
His sources are a thesis by American scientists in the 70s and a book on chess science and Soviet scientists in the 80s, IIRC. It checks out.
The thesis and book were a pain in the ass to find electronically, and I'm not about to do that again now -- I can bring it about later. Although his notes in the book are a start that at least makes his claim plausible:
The definitive study on chess players was carried out by the
physiologist Leroy DuBeck and his graduate student Charlotte Leedy.
They wired up chess players in order to measure their breathing rates,
blood pressure, muscle contractions, and so on, and monitored the
players before, during, and after major tournaments. They found
tripling of breathing rates, muscle contractions, systolic blood
pressures that soared to over 200—exactly the sort of thing seen in
athletes during physical competition.
See the original report, Leedy’s thesis, “The effects of tournament
chess playing on selected physiological responses in players of
varying aspirations and abilities” (Temple University, 1975) or their
brief report (Leedy, C, and DuBeck, L., “Physiological changes during
tournament chess,” Chess Life and Review : 708).
In a telephone conversation, DuBeck also tells the story of the
international match in the early 1970s between grand masters Bent
Larson and Bobby Fischer, in which the former had to be given
antihypertensive medication in the middle of his losing match; his
blood pressure remained elevated for days afterward.
The Kasparov-Karpov report is from the New York Times, 20 December
1990. And for that special chess fan out there who just can’t get enough of this subject, may I suggest as the perfect gift a copy of
Glezerov, V., and Sobol, E., “Hygienic evaluation of the changes in
work capacity of young chess players during training,” Gigiena i
Sanitariia 24 (1987), in the original Russian.