When I took the note from class
my instructor begins with
Ax+By+Cz =1, then solve for A when B,C are 0, and solve for B and C by doing the same.
My question is, where is Ax+By+Cz =1 coming from?
Also, I need some help visualize the boundaries for the integral
Your instructor writes the equation $Ax + By + C z = D$ because this is the general equation for a plane. You can normalize $D$ by dividing through if we know the plane doesn't contain the origin, this is where the $1$ comes from.
$$\frac{A}{D} x + \frac{B}{D} y + \frac{C}{D} z = 1$$
By renaming the constants, we arrive back to the general expression, so we can proceed without loss of generality with the equation your professor wrote.
$$A' x + B' y + C' z = 1$$
where $A' = A/D, B' = B/D$ and $C' = C/D$. In reference to how your professor obtains the bounds, he/she solves for $z$ in terms of $x,y$ i.e $z = f(x,y)$. Therefore $f: D \to \textbf{Plane}$ where $D$ will be the domain you are integrating over. To determine that, you observe that it is just the intersection of your plane with the $x,y$-plane, which in this case is $z = 0$. From here you'll get a triangular region in the plane and now you determine the limits from that picture.