Consider the following line
[FEN ""]
[StartPly "23"]
[StartFlipped "1"]
1. d4 Nf6 2. c4 c5 3. d5 e6 4. Nc3 exd5 5. cxd5 d6 6. e4 g6 7. Bd3 Bg7 8. Nge2 O-O 9. O-O Nbd7 10. f4 a6 11. a4
Stockfish says that c4 is the best move here. The line then proceeds normally, as if nothing happened, with 12. Re8
I know that the general game plan is to expand on the queen side, but I don't see why this pawn sacrifice is so necessary.
What is the idea behind this sacrifice then?
The pawn is only temporarily sacrificed. Black can regain it tactically after a few moves, although it's not readily obvious how that happens. It should be mentioned that instead of 11...c4, 11...Qa5 which prepares c4, is a completely fine move. Here I provide a few sample variations that show what happens if White desperately tries to hold on to the extra pawn.
[FEN ""]
[StartFlipped "1"]
1. d4 Nf6 2. c4 c5 3. d5 e6 4. Nc3 exd5 5. cxd5 d6 6. e4 g6 7. Bd3 Bg7 8. Nge2 O-O 9. O-O Nbd7 10. f4 a6
11. a4 c4 (11... Qa5 {is a fine move, preparing c4. For example} 12. Ra3 c4 {and now the pawn cannot be taken without seriously compromising White's position}
13. Bxc4 $2 Qc5+ 14. Qd4 Nxe4 $1 $15) 12. Bxc4 Re8 {and it is difficult for White to hold on to the extra pawn. For example}
13. Ng3 Nc5 14. Bd3 $6 (14. Re1 $6 h5 15. e5 dxe5 16. fxe5 $2
(16. f5 {is better}) 16... Ng4 17. e6 Bxe6 18. dxe6 Bd4+ $19)
(14. e5 $1 dxe5 15. f5 $1 $14 {is good.})
14... h5 {threatening h4, which undermines the g3 knight that
defends e4. There's not much that White can do about it.}
15. h4 $4 ({Admitting the mistake with} 15. f5 {is better} h4
16. Nh1 Ncxe4 17. fxg6 Qb6+ 18. Nf2 fxg6 $15)
15... Ng4 $19 {and now h4 falls and with it the game as well.} *
The idea behind the sacrifice is to simply mobilize Black's pieces. The Black knight wants to come to c5 and pressure e4. Then the rook can join in from e8 to attack e4 as well. That also makes the knight on f6 more effective, and so on...
