For anyone else out there who may be a chess imbecile like myself... it took me a while to figure out why black king doesn't take white rook:
It's because if they do, white has no legal moves, but king isn't in check. This condition, where a player can't move, but the king currently isn't in check, results in a stalemate (which is considered a draw).
Therefore, white's baiting of black into that position by sacrificing pieces is interesting.
A stalemate [0] occurs when the side to move has no legal moves but is not in check. Note that a stalemate is a special case of a draw: every stalemate is a draw but not every draw is by stalemate. In the final position, if Black captures the rook, it is stalemate. If he does not, White will keep giving check on the seventh rank and the game will soon be a draw by repetition[1].
This game is particularly special because before 46. Bb3 White has six pieces, all of which have legal moves, making stalemate apparently unlikely and checkmate inevitable from Black's aggressively placed Queen and Knights. It's surprising to a human that within a few moves, White is able to force Black to capture or block all 6 of those pieces and give stalemate. There are famous combinations from human play where one side manages to sacrifice "desperado" pieces for stalemate [2][3], but nothing so ingenious as this one.
Specifically, Queen to f4 forces Black to both capture the Queen and block the pawn on f3.
Bishop to b3 is forced - nothing else would stop Queen to d1 and checkmate. Actually, most of the game plays itself beyond 42 or so. It's remarkable that it got into that position, though.
(Have barely played chess since I was a kid, but guessing below until you get a better/accurate answer.)
White was in trouble before throwing away the bishop as the black queen was in a deadly position, so it looked to get any free pieces it had blocked from moving as a viable strategy at aiming for a draw.
As for the stalemate, I assume the 50-move rule, part (a)? It was either going to draw through losing the rook or via 50 moves without losing/taking a piece.
"The game is drawn, upon a correct claim by the player having the move, if (a) he writes on his scoresheet, and declares to the arbiter his intention to make a move which shall result in the last 50 moves having been made by each player without the movement of any pawn and without the capture of any piece, or [snipped]"
It's because if they do, white has no legal moves, but king isn't in check. This condition, where a player can't move, but the king currently isn't in check, results in a stalemate (which is considered a draw).
Therefore, white's baiting of black into that position by sacrificing pieces is interesting.