You can think of $\int \int f(x,y)dxdy$ as "weighted sum" of the region. Your $f(x,y)$ function tells you "how important" each point is. If your weight is $1$, then each point is worth one unit, so the integral just "adds up" the points, and you get the area. If $f(x,y)$ is the height, then the integral is adding up all those heights, so you get the volume.
In calculus, there is a distinction between things that are infinitesimally small, versus things that are zero. The height is not zero, it is an infinitesimal. So instead of taking $\int \int 0dxdy$, we should take $\int \int dxdydz$. This is then the volume of the region. We can factor out the $dz$ and consider this to be $(\int \int dxdy)dz$. The quantity within the parentheses is the area; if we were calculating the volume of a region, we could calculate it by integrating along the $z$-axis and finding the area of each cross section, i.e. $dV =Adz$.
Keep in mind that whenever you calculated the size of a geometric quantity, anything of a larger dimension will be "infinitely" large, while anything of a smaller dimension will be "infinitesimally" small. For instance, if you're calculating areas, then any volume will have an infinite area (it's the sum of infinitely many areas stacked on top of each other), and any line segment will have infinitesimal area. We need to distinguish between a line segment having no area, versus a line segment having no measure at all. When we measure a line segment with cm$^2$, we will get zero, but when we measure it with cm, we get a positive number. And if we want to find the area of a region by integrating a bunch of lengths, then we need to distinguish between these two measures; when we integrate a bunch of lengths, we're adding up a bunch of lengths that each have zero area individually, but together have positive area.
Similarly, by multiplying $dxdy$ by the height, and treating the height as being zero to get the area, you're measuring the region in three dimensions. And measuring area (a two-dimensional quantity) in three dimensional units results in zero.