First off, it helps if you include a link to the source. According to that, Q1 acts as a common-base amplifier and Q2 acts as a common-collector amplifier.
In this circuit, C1 is connected directly to the 5V DC source, which means it is always at 5V. It actually has no influence whatsoever on the circuit. In real life, it's probably a bypass capacitor. In the simulation, you can remove it with no change in behavior.
I tried simulating it in PartSim, but the behavior is rather odd. At the start, Q1 turns on and Q2 stays off. The inductor current (and thus the Q1 collector current) is zero. Q1's base-emitter junction acts like a diode, allowing current to flow from V1 to the resistor. VCE is almost zero, so the inductor voltage is equal to the B-E diode drop.
The inductor current rises at a constant rate. At the same time, the base current drops as Q1 starts acting less like a diode and more like an emitter follower. Their sum (the resistor current) is constant. About a quarter of the way through the cycle, the base current goes negative. (!!)
After a while, Q1 turns off. The inductor voltage polarity flips, raising the voltage of the middle node above V1. The B-C and B-E junctions in Q2 are both forward-biased, with B-E producing R1's current and B-C carrying the rest of the inductor current. With a constant voltage across it (the B-C diode drop), the inductor current falls at a constant rate, eventually going negative. (!!) Eventually, Q2 turns off and Q1 turns on, restarting the cycle. The output is a clamped square wave with an amplitude of about 600mV.
The current waveforms for the two transistors look pretty similar, so I'm skeptical of the source's description. Autistic's cross-coupled oscillator explanation sounds a lot closer, but your circuit is missing the capacitors, only has one inductor, and Q2's collector is AC-grounded. Compare this, for instance:
I don't understand why the inductor current starts flowing through Q1's C-B junction, or why the Q2 B-C/inductor voltage remains constant even after the inductor current reverses. Perhaps the junction capacitance plays a role, somehow? Regardless, it does seem to oscillate. Maybe someone else can figure this out. Here's my transient simulation showing start-up and a full cycle.
Light blue = Q1 collector voltage
Black = output voltage
Light green = Q1 collector current
Orange = Q2 collector current
Purple = Q1 base current
Red = Q2 base current
Yellow = R1 current
Dark green = inductor current (into top pin)