How does double Riemann sum actually work?

Question

I'm in an advanced calculus class and studying double integral. My question is about how double Riemann sum actually work as algebraic steps? I mean I understand essentially it is summing up the volumes of small rectangles and I can make sense from it by thinking it as a 2D-array with 2 counters one of which is for row and another one is for column for indexing the correct base 2D-rectangles. But I get confused each time I try to view it as 2 nested plain summations, that I'm not sure which entity is the summand of which summation. I got that the $f(x_{ij},y_{ij}) \triangle A$ is the summand of the inner summation/sigma, but I started to confuse when I tried to think the result from the inner summation as the summand of the outer summation. This is the expression in my textbook that I'm talking about: $\sum_{i=1}^{n}\sum_{j=1}^{m} f(x_{ij},y_{ij}) \triangle A$. Thanks!!