In principle a ferromagnetic core is not needed to transform voltages through adjacent coils- one can pick up a changing magnetic field in a nearby coil without a core. However due to the magnetic properties of a material like iron (it has high magnetic permeability - the physics of this is interesting but going beyond the scope of this course), it can carry a large magnetic field, which increases the magnetic flux and allows the magnetic field lines to penetrate both coils easily.
BTW there are a number of practical considerations in the engineering of transformers that we are ignoring here in our idealized picture of them.