what does "sacrifice a rook for a pawn"mean?
【Then he awaited Yarborough's pitch. He knew one was coming. Inpolitics no one sacrifices a rook for a pawn. Outside the hotel the i Senator had yielded to Presidential pressure (and a little Irish muscle) Now he was entitled to lay an arm on Larry. O'Brien had written the textbook of Kennedy politics; he knew. 】

1.what does "sacrifice a rook for a pawn"mean?
2.what does "lay an arm on"mean?

Question

what does "sacrifice a rook for a pawn"mean?
【Then he awaited Yarborough's pitch. He knew one was coming. Inpolitics no one sacrifices a rook for a pawn. Outside the hotel the i Senator had yielded to Presidential pressure (and a little Irish muscle) Now he was entitled to lay an arm on Larry. O'Brien had written the textbook of Kennedy politics; he knew. 】

1.what does "sacrifice a rook for a pawn"mean?
2.what does "lay an arm on"mean?

Answer

Rook and pawn are two pieces in chess. Rook is an important piece and you have only two of them. Pawn is less important than rook (you have eight pieces of it).
When you play chess, it is not reasonable to loose a rook for saving a pawn (of course there are some exceptions).

Answer

From chess. Basically, a rook is worth 5 points  while a pawn is worth only one. If you lose a a rook for a pawn, you lose the advantage in the chess game. However, extreme circumstances may make it desirable for you to do so. It would entail a big rist though.