Explain how this astable chain(?) circuit works
This is called a phase shift oscillator where oscillation occurs using 60 of 90 deg of each cap (RC LPF) to achieve the additional 60x3= 180 deg needed to get positive AC feedback.  This is why an odd number of stages is needed.  

This design, however, filters each base current so there is not much variation in LED intensity.  It would be far brighter to move the cap from the base to the collector.

My improvements Here



Since the load is nonlinear LED, the oscillator is not sinusoidal.

The result is actually 2 of 3 LEDs ON with a moving “black hole to the left” as the cascade moves to the right.

5 stages would share 180 deg to shift at 180/5 deg per stage.