Indeed as echoone mentions, this is a generalization of the mean value theorem from single variable calculus.
Consider the rectangle $[x_0,\ x_0 + s]\times[y_0,\ y_0 + t]$. The idea is to apply the mean value theorem component wise to $u(x,y)$.
By taking $x$ fixed, we can consider $u(x,y)$ as a function of $y$. We can then write $u(x_0, y_0 + t) - u(x_0, y_0) = tu_y(x_0, y_0 + t^*)$ Now we apply the mean value theorem again to $x$ $u(x_0 + s, y_0 + t) - u(x_0, y_0 + t) = su_x(x_0 + s^*, y_0 + t)$ Combining the two yields the desired expression. There appears to be a small error in that the statement should read $u(x_0 + s, y_0+t) - u(x_0, y_0) = su_x(x_0 + s^*, y_0 + t) - tu_y(x_0, y_0 + t^*)$ and not $u(x_0 + s, y_0+t) - u(x_0, y_0) = su_x(x_0 + s^*, y_0 + t^*) - tu_y(x_0, y_0 + t^*)$ The $t$ in the first term on the right is not $t^*$.