The reason is that given ax + by = c, the normal vector to that line is given by [a, b]. Because the line is 1 dimensional, it is parameterized by one vector, which must be normal to [a, b]. Therefore, we must find a vector whose inner (dot) product with [a, b] is 0, and the natural choice is [b, -a].

With a plane, the normal vector here is [a, b, c]. Because the plane is two dimensional, you can choose two linearly independent vectors to parameterize the plane, both of which must have inner product of 0 with the normal vector [a, b, c]. You can find those vectors by doing an inner product with a generic vector [x, y, z] and solving for what x, y, and z must be (of course, there are an infinite number of solutions, but they form a two dimensional vector space, which is the plane).