This is a Hartley oscillator, an oscillator that uses an LC circuit combined with a transistor for feedback.
With the transistor removed, the capacitor and two inductors form a resonant circuit, like the LRC example. Current moves back and forth as the capacitor charges and discharges through the inductors. The transistor amplifies this oscillation and prevents it from dying out.
The transistor cannot conduct until its base is at about 680mV. When the transistor is off, the output is around 4.7 V. Current from the 1k resistor gets a current going to ground through L2. As the bottom of L2 reaches ground, L2 resists any change in current and current continues to flow, now through the capacitor and L1. This brings the transistor base voltage up to where the transistor can conduct, bringing the output low.
Once the base voltage is high, the current through L1 and L2 begins to reverse, draining the capacitor and bringing the base voltage down again, turning off the transistor and bringing the output high again.
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