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1.  Current flows Q1(eb) to ground through R1which slows it down a little to keep from overloading.

2.  This opens Q1(ec), so current starts flowing to Q2(be) through R2 which slows it down a little, again to keep it from overheating.

3.  This opens Q2(ce) so current starts flowing through L1 which builds up a field around L1, and current also flows through C1 and R1 to ground.

4.  When C1 charges up, the voltage at Q1(b) is about the same as at Q1(e), so the current stops flowing through Q1(ec).

5.  This breaks the current flow through Q2(ce), and through L1.

6.  The L1 field collapses and wants to go to ground.  The only exit path as long as the capacitor is charged, is for it to jump the voltage gap across the led.  The capacitor discharges to ground through R1.

7.  The cycle repeats.

Notes:

a.  The time it takes for C1 to charge determines the operating frequency.
b.  R1 and R2 prevent the transistor switches from overloading.  R1 slows the discharge of C1.
c.  The energy to light the led comes primarily from the L1 field collapse when the circuit opens.

Scale up proposals for consideration:

1.  Add more leds in series, and increase the size of L1.  Maybe 100 uH per led.
2.  Leave C1 at 680 pF unless more time is needed to charge L1.
3.  Leave R1 and R2 at 10k and 3.5k.
4.  Maybe add a capacitor in parallel with the leds to smooth L1 discharge pulses.