$f_x(1,1)$ is the slope at $(1,1,f(1))$ of the cross-section of the graph of $f$ in the $x$-direction.
This is the slope of the red line in the figure.

$f_y(1,1)$ is the slope at $(1,1,f(1))$ of the cross-section of the graph of $f$ in the $y$-direction.
This is the slope of the red line in the figure.