Another aspect of the phrase you provided is how English uses "up" and "down". A player is considered "up" if they have an advantage. Likewise, being "down" mean a player has the worse position or piece count.
Because rooks are generally more valuable than pawns, whichever player still has the rook is in a better position and would be considered "up a rook for a pawn".
Combining these two ideas, "white is down a rook for a pawn" means that white has the worse end of the trade involving a rook and pawn. This means that white has lost a rook and only captured a pawn as compensation.