Cartridge starters are common on Jet engines too for cases where you need to get them started ASAP (i.e. military).
Fun fact: most airplanes have a smaller turbine used to start their larger ones. The ones that don't use a mobile turbine cart for the same purpose. It takes a lot of energy to get them started.
A simple (read: I don't understand it better yet) explanation is this: A starter cartridge was placed in the turbopumps of the engine, that ignited and spun up the turbopumps (technically, the turbine of the TPA which then turns the pumps), which then fed fuel and oxidizer into a) the gas generator to sustain operation, and b) into the combustion chambers. The cartridges burned for about a second, and you can hear the screeching sound they made in this launch video [1]
[1] https://youtu.be/E87deQMLHoQ?t=185