Let $a $ be a real. then

$$f (x,y)=\int_a^x g (t)dt-\int_a^y g (t)dt $$

$$=G (x)-G (y) $$ with $$G(X)=\int_a^X g (t)dt $$ and $$G'(X)=g (X) $$ since $g $ is continuous at $\mathbb R $.

thus

$$f_x (x,y)=G'(x)=g (x) $$ $$f_y (x,y)=-G'(y)=-g (y) $$

"Leibniz's formula" is a generalization of the "Fundamental Theorem of Calculus": $$\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x, t)dt= f(x, \beta(x))- f(x, \alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f(x, t)}{\partial x} dt$$

Here, $f(x,t)$ is a function of t only, the upper bound on the integral is just $x$ and the lower bound just $y$. So the derivative of the integral with respect to $x$ is $g(x)$ and the derivative with respect to y is $-g(y)$.