There're some notes on this game on Wikipedia.
I think the key point is this: if Bryne takes the knight, he's going into a cheerless endgame after 18...Qxc5!. He'd be down material and have positional weaknesses to boot. The best he can hope for is a draw; against a player of equal caliber, his position is losing.
Therefore he goes for the complexities of ...
I would not normally answer this question since the other answer points out the basis for the first move of the variation, but only the first move. It left out why white is so lost in that variation, and why Byrne did not play it.
At the end of both lines below, there are written notes explaining the resulting position.
[Event "New York Rosenwald-03"]