Help me understand easy (not for me) concepts in volume integral 
Keep looking at the page for an hour.
Still not sure how I can get the sloping surface of $x+y+z=1$ and integration ranges for $x, y, z$. and why their range is different too.
The book keeps throwing things at me without much explanation. Help me.

This is an attempt to capture how I tend to visualize a region when setting up a volume integral. Consider the following plots:

On the left, the forest of arrows join the $y,z$-plane (i.e. the plane $x=0$) and the given plane $x+y+z=1$. They represent the range that the variable $x$ can take on, $0 \le x \le 1 - y - z$.
On the right, there's a similar array of arrows that join the $z$-axis (i.e. the line $x=y=0$) to the line where the plane $x+y+z=1$ intersects $x=0$. This line has equation $y + z = 1$, or $y = 1 - z$. These arrows represent the range that $y$ can take on, $0 \le y \le 1 - z$.
It's not pictured, but the last step is recognizing that $z$ must range between $0$ and $1$.
Then the volume of the region is given by the triple integral
$$\int_0^1 \int_0^{1-z} \int_0^{1-y-z} dx \, dy \, dz$$
