How does a moving magnet produce current in a coil with no battery?

This is electromagnetic induction, discovered by Faraday. A coil sits in the magnetic field of the magnet, and the amount of magnetic field passing through the coil is called the magnetic flux. When you move the magnet, the flux through the coil changes. Faraday's law says that a changing magnetic flux induces an emf (a voltage) in the coil, and if the coil forms a closed loop, this emf drives a current. The energy does not come from nowhere: it comes from the muscular work you do pushing the magnet against a force that opposes your motion (Lenz's law). If you hold the magnet still, the flux stops changing, the induced emf becomes zero, and the galvanometer reads zero even though the magnet is right there. Motion, or any change in flux, is essential.