White resigned simply because the c3 knight is poison due to a mate in 3 starting with Re1.
Therefore black can just play hxg4 hxg4 then undermine the e5 knight with g5. Notice that the e5 knight gets trapped. There is no way out of both these mating conundrums and these knight trap schenanigans (schevenigen lol!)
Here is the diagram if we give White a free move. That's right, Black had the move in this game and he will smash with hxg4 followed by trapping the knight on e5 (after g5), but even if White moves first, Black is still smashing!!!
[FEN "8/ppk3p1/1r6/3pN2p/2p2PP1/2n1r2P/P2R4/K1R5 w - - 0 0"]
1.Rxc3 (1.Rdc2 (1.gxh5 Re2 2.Rxe2 Nxe2 3.Rc2 Nxf4 4.h4 Kd6) 1...Ne2 2.Rf1 (2.Re1? Nd4) h4
3.g5 Re4) 1...Re1+ 2.Rd1 Rxd1+ 3.Rc1 Rxc1#
By the way, I checked this position with a computer (Stockfish 8, depth 28) just to make sure I was right, and the initial position here before Rxc3 scores as -5.4 (if Black moves as in the game) and -3.2 if White moves here. Now here is the position if it is Black to move, as in the game.
[FEN "8/ppk3p1/1r6/3pN2p/2p2PP1/2n1r2P/P2R4/K1R5 b - - 0 0"]
1...hxg4 2.Nxg4 (2.hxg4 g5 3.Nf7 (3.Nxc4 dxc4 4.Rdc2 gxf4 5.Rxc3 Rxc3 6.Rxc3 Kd6
7.Rxc4 Ke5 8.Ra4 f3) 3...gxf4 4.Ng5 Rg6) 2...Re2 3.Rxe2 Nxe2 4.Re1 Nxf4 5.Rh1
(5.Rf1 Nxh3) 5...c3 6.a3 d4
If White doesn't play right, then Black just immediately wins material or mates. If White plays right, then Black gets a winning rook endgame because of his massive advantage in pawns.
At the end of the day, he probably resigned 20 or so minutes after making that move just based off the endgame. Remember, Mikhail Botvinnik was like the most heavy duty Russian chess dude out there.
Chess was serious business to him and the Russians trained in endgames more than other people did. Botvinnik actually tackled a line of the Semi-Slav (a line of the Semi-Slav Botvinnik I think??) for 100 hours worth of preparation! These top players gave Chess their 110%.