The easiest way I've seen is to just remember that the integral is the limit as all the dx's go to zero and a sum over them goes to infinity
so your integral over f(x,y,z)dxdydz is a sum over all the little cubes of size dxdydz
Okay, so you can think of going from a single integral to double integral as instead of adding squares adding cubes from a limit. For instance a single integral should be 2D while a double integral is 3D. That made it a lot easier to understand. Thank you.
When you are doing a single integral, you are summing over little lengths dx, you're adding up the value of f[x] multiplied by the small length dx (this gives you the area of a rectangle over the length dx)
When you do a double integral you are summing over little flat squares of sides dxdy, this can be taken to be a volume, just as the single integral can be taken to be an area, if you make your f(x,y) be the 'height' then you are summing over little parallelepiped (I hope I spelled that correctly) of height f(x,y) and base dx dy.
When you take a triple integral you are summing over little cubes of sides dxdydz, this again can be interprited as a 'volume' integral for some higher dimension, you could take the mass to be the 'volume' under the graph of the density function.
But human brains aren't wired to let us think about anything other than 3 dimensions.
I hope you understand what I mean and I haven't just made it seem more complicated :p