Translation property of 2-D discrete Fourier transform
Very informally, they use the property $e^{j\pi} = -1$ (which is correct), the property $e^{ab}={(e^{a})}^b$ (valid for certain values of $b$), and derive the final expression for $b=x+y$ when $x+y$ is an integer.

The pattern $(-1)^{x+y}$ looks like a checkerboard.