How does a wireless charger work without a magnetic core passing through both "coils"?
the alternating current in the primary
  coil induces alternating
  electromagnetic field in the core and
  that field induces current in the
  secondary coil


Yes, but "in the core" is not necessary.  A coil creates a magnetic field all around it, whether there's a core there or not.  The core just confines the magnetic field so that less of it "leaks", which makes the transformer more efficient.

You can also make the transmission more efficient by putting a capacitor in series with both coils so that they resonate at the same frequency:  http://en.wikipedia.org/wiki/Resonant_inductive_coupling