Here is a screenshot from the video of the critical moments of Peter Eastgate' path to winning, at 14:18, the moment the six of diamonds was drawn.
According to PokerStar's narration, this was the only possible card that could secure Eastgate a win, and it was drawn against all odds--interesting stuff. However, I do not understand why this made Eastgate win or why it was his only chance.
From what I can tell, 'No Limit' just means there are no restrictions on betting. Otherwise, it is standard Texas Hold 'Em in which aces score the highest.
In this case, the best hand for Montgomery was a three kind of kind aces against Eastgate's three of a kind of sixes. As a tie for the same kind of hand, aces score higher.
What am I missing here in the rules of Texas Hold 'Em?