Holy crap, they actually bloody did it. They caught a booster. First time attempt! Incredible engineering. Small fire but that can be resolved right? 
Me and my brother were discussing this. In order to run a rocket motor you need a turbo pump. Turbo pumps burn either fuel rich or oxydizer rich. Looking at the footage I believe the raptors in use during 'landing' share a turbo pump or a turbo pump exhaust. You can see when the engines back off the turbo pump is spewing unburnt fuel everywhere in a big plume of yellow flame (O2 starved gas burn).
In order to light, extinguish, relight, throttle etc those engines the turbo pump must be kept running, no? You won't see it on the way up as (apparently) they are normally "closed cycle" and the turbo pump exhaust is reinjected and not vented.
I may have to go and rewatch all the techie detail videos on the raptors again.
I don't think that's strictly required.
The fuel pumps are probably electrically started and then run on combustion energy. These run on partially burned fuel - oxygen-rich methane or methane-rich oxygen gas output. This partially combusted fuel is then sent to the main combustion chamber where it is fully combusted to provide thrust.
The ignition is via spark ignition so does not require any pump to operate.
So turn electric fuel/LOX pump on, engage igniters, igniters cause fuel combustion which causes turbos to spool. All it would need is the minimum amount of fuel/LOX to get the reaction going. Some very rapid control electronics and software will be monitoring the fuel/LOX rates to ensure that the engine starts up quickly as I am sure it is possible to put it into very dangerous and failure-prone operating modes.
Shutdown would be by isolating fuel/LOX which will result in engine running fuel rich as methane is denser than oxygen. This results in the visible dark orange flames of poor combustion before shutdown. They might keep methane going for a tad longer to avoid overheating the nozzle, as that is fuel cooled, but just a guess on my part.
As can be seen here, the startup and shutdown are pretty rapid.
SpaceX relies on this since the three Raptors aren't enough to slow down until the terminal phase of the flight so they have to alternate between 13 and 3 raptors on at various stages. On Falcon 9 a single Merlin engine is actually sufficient to allow the empty Falcon 9 to lift off again so the descent profile is based on very careful timing of the single engine relight. Merlin uses a consumable liquid ignition fuel and takes longer to ignite than Raptor, but not exactly sure on why.