How do moving charges produce magnetic fields?

I'm tutoring high school students. I've always taught them that: A charged particle moving without acceleration produces an electric as well as a magnetic field. It produces an electric field because it's a charge particle. But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field. Why? What happens to it when it starts moving? What makes it produce a magnetic field when it starts moving?