I am confused only on one small part. In the image below, he writes
$$\frac{-\hbar ^2}{2m}\frac{d^2X}{dx^2}=E_xX; ...$$
But shouldn't the right hand side of each equation be
$$...=E_i(XYZ)$$?? Because in the original equation the right hand side is $...=E\psi=(E_x+E_y+E_z)(XYZ)$. I see how he he takes each component separately, but how does he break up the product $XYZ$??
As you have observed, if we try to solve the nondimensional $-\nabla^2 \psi(x,y,z) = E\psi,$ then letting $\psi = X(x)Y(y)Z(z)$ leaves you with $$ -\nabla^2 \psi(x,y,z) = -(X''(x)YZ + XY''(y)Z + XYZ''(z)) = E X(x)Y(y)Z(z).$$ Dividing by $XYZ,$ $$ -\frac{X''(x)}{X(x)} - \frac{Y''(y)}{Y(y)} - \frac{Z''(z)}{Z(z)} = E_x + E_y + E_z.$$ So we have that each term on the left hand side depends on only one coordinate and therefore is completely independent from the other terms. The only way to satisfy this equation for any combination of $(x,y,z)$ is to require that three 1D equations be satisfied, namely $$\frac{-X''(x)}{X(x)} = E_x$$ and same with the other coordinates. OR, leave $E$ as is, and do a little algebra, $$ -\frac{X''(x)}{X(x)} - E = \frac{Y''(y)}{Y(y)} + \frac{Z''(z)}{Z(z)}.$$
Ah so we see that the left hand side is only a function of $x$ and the RHS is only a function of $y$ and $z$. Setting the right hand side equal to an arbitrary constant, say $$-\frac{X''(x)}{X(x)} - E = E_x - E,$$ One obtains the same equation as previously, in each coordinate.
A perhaps better way to look at this problem is from a more physics-y perspective. Seeing that the problem to be solved is invariant when you permute ($X$ and $Y$) or ($X$ and $Z$) or ($Y$ and $Z$), one concludes that due to the symmetry of the problem that the same equation must be solved in each coordinate. This symmetry arises mathematically because when you do separation of variables, you get 3 independent equations. Of course boundary conditions have been ignored in this argument, but they can be accounted for pretty easily for the typical model problems, such as "particle in a box with different side lengths".