If we color the cells of the board like those of a chessboard (alternatively black and white) then the sum of the numbers in the white cells must be equal to that of the numbers in the black cells. The necessity is clear: in each move, you make equal contributions to both sums (white and black), and if in the end you reach $0$, i.e. both sums equal to $0$, then they must have been equal in the beginning as well. If, conversely, the two sums are equal, we can clear the board one column at a time. When we're left with only one column, we continue clearing up cells one by one until we're left with a $2\times 1$ board. By the hypothesis (white and black sums are equal) the two numbers we're left with are equal and we can finish.

Why the black sum must be equal to the white sum? Thanks