PCB - Why to use Iron chloride?
This link has the answer (and also tells you that ferric chloride is not the best choice as it's not reusable)

The core answer is that it performs a redox reaction:


  2 Fe(3+) + Cu(0) –> 2 Fe(2+) + Cu(2+)


That turns copper metal on the exposed PCB into copper ions that dissolve in the etchant, removing copper from areas of the PCB not covered by the "mask" that was applied photographically earlier in the process.