For every grid size, player 1 has a winning strategy.
Suppose we play Chomp on an m by n grid, where m and n are at least 1, and
m and n are not both 1. Suppose player 2 has a winning strategy. We use
player 2's winning strategy to construct a winning strategy for player 1.
Suppose player 1 picks the block of chocolate in the lower right corner.
How would player 2's winning strategy respond? Say player 2 responds by
choosing block X.
Then player 1 has a winning strategy by "stealing" this response strategy!
Player 1 can simply choose block X for its first move, and then follow
player 2's winning strategy to win the game.
Since exactly one player must have a winning strategy, this strategy-stealing
argument shows that it must be player 1 who has the winning strategy. But the
argument does not make it clear what the winning strategy is!