In this context, deriving the plane equations does not refer to the equations themselves, but to the coefficients $A$, $B$, $C$ and $D$.
The equations are simply plane equations for a plane defined by three points, where the first plane is defined by the three points $(x,s,t)$, the second plane is defined by $(y,s,t)$ and the third plane by $(z,s,t)$. They can be seen as auxiliary objects in deriving equations for $x$, $y$ and $z$ in terms of $s$ and $t$.
Side note: In general it is not possible to derive plane equations as it is done here, even if the triangle is non-degenerate in $(s,t)$-space and in $(x,y,z)$-space. Just because the points $x,y,z$ are non-collinear and $s,t$ are non-collinear does not mean that for example the three points $x,s,t$ are non-collinear.