How does the magnet exert a force on the metal coil without touching?

A magnetic field and an electric conductor, such as a coil of wire, moving relative to one another. Current will flow whenever the conductor crosses lines of force with the magnetic field The faster the magnet or coil moves (or more coils/more current), the greater the amount of current that is produced.

A magnet is any body which produces a magnetic field and exerts a magnetic force. In a magnet, the domains are aligned in a single direction. This creates a field around it. Magnetic field lines represent the region where the magnet exerts a magnetic force. Like poles repel each other while unlike poles attract each other.

A magnet exerts a force on metal coil by inducing magnetism. Under the influence of the magnetic field, some of the domains of the metal coil also align in a single direction making it act like a magnet. Thus, the magnet exerts force on it.