Can anyone give me a good explanation of why the plane Ax+By+Cz=D has normal vector (A,B,C)? It just seems so random.
Rid yourself of any such notion. Nothing such as that is ever random in mathematics.
What you must understand is that the normal is perpendicular to ever line contained in the plane.
That is the dot product of the normal with any vector determined by two points in the plane is equal to zero.
The usual form of a plane is $\displaystyle n_1 \left( {x - x_0 } \right) + n_2 \left( {y - y_0 } \right) + n_3 \left( {z - z_0 } \right) = 0$: where $\displaystyle \left( {x_0 ,y_0 ,z_0 } \right)$ is a point in the plane and the normal is $\displaystyle N = \left\langle {n_1 ,n_2 ,n_3 } \right\rangle $.