It's mostly because the working fluid is at very high pressure (much denser than the air coming through an aero engine). The turbine side works with hot gas at 100+ bar, and the pump side is dealing with liquid. Therefore small parts can exert a lot of force (large pressure differences) and do a lot of work (high speeds).
The most similar technology is a boiler feed pump (used to feed the boilers in a steam turbine system, e.g. in a ship or power station) - these work in a similar way, high pressure steam is bled off from the system to drive a small turbine, which drives a high pressure pump to feed the boiler, which feeds the 'main' steam turbine. Similar pressure (>100 bar), but the turbine on the rocket engine needs to handle much higher temperatures (hot gas rather than steam).
The most similar technology is a boiler feed pump (used to feed the boilers in a steam turbine system, e.g. in a ship or power station) - these work in a similar way, high pressure steam is bled off from the system to drive a small turbine, which drives a high pressure pump to feed the boiler, which feeds the 'main' steam turbine. Similar pressure (>100 bar), but the turbine on the rocket engine needs to handle much higher temperatures (hot gas rather than steam).