How does Rühmkorff induction coil start and periodically produce sparks?
It's pretty straight forward:

Spring terminal T1 and adjustable terminal T2 are adjusted so that they are in contact when no current flows.

When you connect the battery to the device, current flows. 

This causes a magnetic field in "soft iron core 1."  

That pulls on "soft iron pole," causing it to move.  This opens the connection between the two terminals.

When the connection between the two terminals is opened, the magnetic field collapses, and the soft iron pole moves back to its resting position.

The circuit is now connected again, and the cycle repeats.

All of this happens in the primary coil of the device.

With the primary pulsing on and off, a pulsing high voltage is generated in the secondary coil.



The ignition coil on the old Ford model T worked the same way.

It used a vibration system to generate high voltage constantly.