Why does rotating a black hole induce a magnetic field?

It's been a while but IIRC there are 4 classes of theorized black hole that satisfy the Einstein-Maxwell EQ's in a flat spacetime. You've got static holes (Schwarzschild), rotating ones, static WITH CHARGE, and rotating WITH CHARGE. The last one (rotating WITH CHARGE) will have its own magnetic field, even without surrounding ions to whip around to generate one. The rest won't bring about their own magnetic fields (as far as I am aware), and rely on whipping around surrounding ions to generate them.

Generally most black holes have negligible charge (as far as I'm aware) and said charge is often ignored in physical calculations.

If the black hole isn't one that rotates with a non-negligible charge, and generally that isn't even considered, but has a magnetic field that is notable, it's most likely due to plasma and other ion sources rotating around the black hole (see: hot as heck accretion discs, etc, etc).

Note that when a charge moves, it generates a magnetic field.