This was done by having cameras placed over each board and each camera had its own monitor in a control room. (The excuse being that lots of shots were needed for the show.) The monitors were paired so as to display who was playing who.
Derren Brown had two tricks. One was to be able to keep a straight face and appear confident the other was to hide an earpeice in his ear. The helper in the control room simply observed the monitors and called out the moves. The only difficulty was in getting Derren Brown to understand the description of the moves... having to fall back on imprecise descriptions...move the one that looks like a castle two squares to the left etc. etc. And have him fumble with a piece in his hand while he workd out where g5 was...and then do a retake so it would look slick on the telly.
As for the extra player...the helper in the control room solved this with pocket Fritz or something similar.
There was some other basic trick to do with the numbers at the end..and that's it. No mystery.
Although the players weren't in on it ... you have to reckon they would have been able to work it out...