The technique is revealed at the end of the video. 
Sweet video. I figured out how he did it before he explained it, though. 
OK but did you figure out how he knew in advance the number of pieces remaining each opponent would have at the end?
– Hal Terrie
I quickly figured out how he did it, but I never figured out how he knew in advance the number of pieces remaining for each opponent.
They always say to disguise the lie with a grain of truth. He said he’d studied all the players’ games. As we know, high-level chess players play a lot of chess and record an enormous number of their games. Lots of data means a good sample size for a statistical analysis. Also, since high-level games end by resignation more often than by checkmate, each player probably has his own mental “trigger condition” that tells him when to resign. It’s possible that Brown simply looked at each player’s games and averaged out how many pieces he and his opponent ended each game with, perhaps throwing in modifiers for the matchups he knew he was going to make. Note that in one game, he was a piece off – an inexact prediction but close enough to be explained by statistical error.