I would probably have to say no. The fats in the brain are from omega 3 DHA's. The only way to get them would be fish or plant foods. The fats in meat are saturated. Your brain can make the cells from saturated but they are more rigid and don't absorb nutrients as well. If there is any basis to their story I think it would make more sense maybe that our ancestors started to eat fish?
Also chimpanzees currently cannibalize each other and kill other animals to eat. I don't see why they wouldn't have in the past. http://www.youtube.com/watch?v=WDFh5JdYh7I
(a little graphic its chimps hunting a monkey)
I think there are a few different ways evolutionary changes happen. Natural selection. Random mutation could just be a mutation of our genes to give us larger brains. The mutation could also have given us smaller brains, hence, random. Also attraction could have swayed to more intelligent partners so the less intelligent wouldn't have had a chance to mate. We don't really control what we are biologically attracted to. It's written in our DNA from the moment of inception. Actually considering the diversity of life I would say there are a myriad of reasons and not just 1 to explain it.
I wouldn't care either way whether we evolved to get larger brains from meat. If it was true I would just have to accept it, but if there are no facts to back it up I would just have to chalk it up as myth.
I just basically know what I know from a little bit of research and countless hours of animal planet. It would be best to talk to an evolutionary biologist if you wanted the whole story.