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 
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 rubber cord, a single felt boot (since most propellers were two blade), two-three men: one holds the propeller, others pull the cord. When they can't pull it any more, propeller is released.
Main problem was to warm up the engine in winter. For liquid-cooled ones that was done by repeatedly draining the system and refilling with hot water. I don't remember what they did with air cooled ones, maybe a blower of some kind, or just a blowtorch to the cylinders. There aren't many options on a snow field out somewhere in nowhere.
The original, not the remake.
> The starter became famous as a plot device in the 1965 movie The Flight of the Phoenix, in which pilot Frank Towns (James Stewart) has a limited number of cartridges with which to start the makeshift aircraft's engine; this was also featured in the 2004 remake of the film.
But not all of them.
I wonder how tight the tolerances were on these engines?
When the U.S.A. entered WW2 it was Ford that tackled the problem of making aero engines en-masse. Ford took Rolls Royce designs that had been used for the war thus far (the U.S.A. being a late entrant) and realised that the blueprints provided were not fit for their purposes.
Until then Rolls Royce had been making each and every engine in a fairly bespoke way with the parts not exactly interchangeable between engines. Tolerances varied so fettling was always required. Ford fixed this and delivered precision engineering for a much better product.
Were the Rolls Royce versions of the engine the ones that needed the Coffman system with the superior Ford versions not requiring it?
That is my guess.
This was not the end of really-difficult-to-start engines though. Motorsport took it to a whole new level in the post war years.
Aero style supercharging didn't last for very long in the pinnacle of motorsport - Formula 1 - due to the thirstiness of the engines. Naturally aspirated engines (with the Ford Cosworth DFV) ruled the roost for many decades and then Renault came along with the turbo. This changed everything. However, to get the turbo to work precision engineering had to solve the problem of extremely high pressure gasses wanting to escape past the piston rings.
To fix this the pistons were made 'larger' than the cylinders. So at room temperature, with a cold start, there was no way for mere mortals to hand crank the engine. A very big starter motor in the garage was needed and only at operational temperatures with a little bit of thermal expansion could the engine run freely.
The flywheel was very light on these engines so only true drivers could keep these things from stalling, and if stalled whilst on the track, that was that, finito.
BMW made the most legendary turbo era engines, 1400 horsepowers in qualifying trim, with the engine having to be rebuilt after a handful of laps. The blocks for these engines came from their 4 cylinder road cars. According to legend the key ingredient for the strength of these engine blocks was urine and cold outdoor weather.
Wikipedia only lists the Merlin 32 as using a Coffman starter, as it went into naval aircraft maybe there was a requirement to be able to restart an engine in flight.
...beginning in 1939.
Ford - Manchester UK only got going in 1940.
I didn't realise that Ford in America turned down the job, that must have been Mr very contrary Ford for you!
I must have gleaned that 'fact' regarding the Rolls Royce blueprints being no good for Ford from the History Channel. I do wonder though about how much Ford thinking influenced Rolls Royce and their shadow factories elsewhere in England, where they did use relatively untrained staff to crank out the volume.