I know that the phenomenon of a magnetic field being generated by a current running through a wore is because of relativistic effects in the electric field that creates magnetic effects. But how exactly does a magnet relatively spinning around a coil of wire generate an electric current? (Sorry if it's a stupid question)
Faraday's law of induction tells us that a changing magnetic field induces an electric field (a voltage), and this electric field will cause a current in a conductor, if the conductor forms a circuit. Chapter 5 of Electricity and Magnetism by Purcell and Morin explains these relations in terms of special relativity.
Magnetic fields push and pull electrons inside the wire. When the electrons move a current is created. Metals do this very well because the atoms are missing valence electrons making it easier for electrons to move.