I don't know how to explain it without a blackboard. Here is a (poor) try.
One has to keep constantly in mind is that $x$, $y$, and $z$ must always be $\ge 0$.
One should also try to visualize the region. It is pyramidal, with triangular faces.
The $z$ is easiest. Since our plane has equation $2x+2y+z=4$, the variable $z$ goes from $0$ to $4-2x-2y$, as long as $4-2x-2y$ is positive. But that will be taken care of later.
Now that $z$ has been taken care of, we need to integrate over the base (in the $x$-$y$ plane) of the solid. So our problem is comfortably $2$-dimensional. Let's deal with $y$. As usual, we need $y \ge 0$. At the top, we need to make sure $4-2x-2y\ge 0$. So we need $2y \le 4-2x$, or equivalently $y\le 2-x$.
Finally, we need $2-x \ge 0$. So $x$ goes from $0$ to 2$.