The object of the cartridge start was to allow the aircraft to be totally independant of ground services (electrics, air or hydraulics)
The cartridge case was filled with a slow burning charge of cordite - not the fast-burn charge of a projectile.
When ignited, the high pressure of the burn was directed to a turbine (about six inches in diameter) which wizzed round at about 50,000 rpm. This was connected to the engine compressor via a reduction gear and gearbox and retractable coupling (not unlike a car "bendix" drive)
Since the burn lasted about 5 seconds, it spun the engine up to a reasonable rpm and as long as fuel was delivered to the combustion chambers and the high-energy igniters worked OK, the engine started.
The early starters only had a single cartridge but later ones had three, so three attempts could be made to start without a re-load being necessary.
On later marks of Hunters the "cartridge" was replaced by a combustion chamber in which IPN (Iso-Propyl-Nitrate (liquid rocket fuel) was burnt.
Both - in my experience - were effective and reliable systems, although when the IPN one went wrong, it usually did so quite dramatically !
Interestingly, does the CAA allow these start systems on civil registered aircraft ? (I believe they ban Coffman starters on Chipmunks ?)