The exchange is the value difference between a rook and a minor piece; winning the exchange means trading a bishop or knight (worth ~3) for a rook (worth ~5).
Because a rook is worth about five points and a bishop or knight about three, swapping a minor piece for an enemy rook leaves you 'up the exchange' — roughly a two-pawn material advantage. The player who gave the rook is said to be 'down the exchange'.
Being up the exchange is a real but moderate edge. The extra rook tends to shine in open positions, on open files, and in endgames where it can attack pawns from the side; the side with the minor piece does best when the position is blocked or the bishop has good diagonals and targets.
Don't confuse 'the exchange' with an ordinary 'exchange' of equal pieces — here the term specifically refers to the rook-for-minor-piece imbalance.
About two pawns. A rook (~5 points) for a minor piece (~3 points) is a meaningful but not overwhelming advantage, and its real value depends on how open the position is.
When the minor piece has strong squares and targets and the position is closed, so the extra rook can't find open files. Good pawns or a powerful bishop can fully offset the missing exchange.
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