Since the case of a plane passing through the origin is trivial, consider a plane that does not pass through the origin.
Let P be any point on the plane.
If there were a line in the direction of the unit normal vector passing through the origin, it would intersect the plane at one point, and it would intersect it at right angles. Call this point A.
It follows that OA is perpendicular to AP, so OAP is a right-angled triangle with a right-angle at A.
The cosine of the angle θ at O is equal to OA/OP. Therefore OPcos(θ) = OA.
Now, the normal vector is either in the same direction as the vector OA, or in the opposite direction. If it is in the same direction, then the dot product between the normal vector and the position vector OP is equal to |OP|cos(θ), which is equal to |OA| as we saw. If it is in the opposite direction, then the dot product is |OP|cos(π-θ) = -|OP|cos(θ) = -|OA|. In either case, the value is always the same for the same normal vector.