I'm trying to understand why $\iint dA$ is the area of the region A.
I thought double integration can be interpreted as the volume under the curve on the area A (as long as the function is non-negative).
But why is $\iint dA$ the area of region A rather than the volume under $\ f(x) = 1 $ on the region A?
When I do the calculation, I know that it's the area of region A but I don't understand perfectly.
You can think of this integral as the volume of a prism of uniform height $1$ above the base $A$. The volume of a prism is just the product of the height and the area of the base.
If you wrote this out formally with units of length for the three axes then the integral would have units $\text{length}^3$ while the area of $A$ would have units $\text{length}^2$, so they would not be exactly the same thing.