Triple integrals can be confusing but apart from the mathematical substitutions for limits and integrands, the best advice I can give is to draw a diagram for each substitution. In a 2D integral you have a plane and in a 3D integral you have a volume.
However you can also look at each 2D integral (there will be 3 pairs XY XZ and YZ) and use this to visualize what is going on each pair and then use that intuition to see what is happening in the 3D space. (This is also useful if you have to get intuition for 4D or higher space - You just slice up the components and integrate them together as intuitively as you can).
Do you want us here to help you with the algebraic derivation or the intuitive geometric concepts?