Turbofan engine 2 spool - Turbine drives the compressor or Starter drives the compressor
The starter drives the entire HP spool (HP compressor, interconnecting shaft, and HP turbine).  The gearbox in question is the accessory gearbox that runs the various pumps and is driven off the HP spool shaft, which by happy coincidence is on the outside of the LP shaft.  The starter is mounted on that gearbox and is able to back drive the HP spool through the gear train for engine starting. 

The LP spool, which is the fan and LP turbine with its shaft running down the center of the HP spool, is just going along for the ride during the start.