Say a plane P has a given equation ax+by+cz=d. Given a point $(x_0, y_0, z_0)$ that is not included in P. When $(x_0, y_0, z_0)$ is plugged into $f(x,y,z)=ax+by+cz-d$ and it outputs some nonzero $e$.
What does $e$ represent?
What information does this give us about $(x_0, y_0, z_0)$ in relation to P?
I believe that the set of all points such that $f(x,y,z)=e$ form a plane with equation $ax+by+cz=d+e$. Is this all that's useful about the extra information? The corresponding parallel plane that the point $(x_0, y_0, z_0)$ belongs to?
You are right that $ax+by+cd=d+e$ describes another plane. More precisely a plane parallel to the first one.
The value of $e$ is related to the distance between these two planes (or between the point and the plane in the first part of the question). More precisely, this distance equals $\frac{|e|}{\sqrt{ a^2+b^2+c^2}}$. The sign of $e$ determines the side of the given plane. With this point of view, the points in the given plane are simply those points having distance $0$ from the plane :)