The circuit is a high gain amplifier.
If it were drawn with a proper layout, it would be easy to see how the circuit works.
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The circuit does not turn on until a resistance is created at the terminals.
When this happens, the voltage on one end of the capacitor rises and this raises the other end.
This creates a current through the capacitor and the same current flows through the base of the second transistor.
This causes both transistors to turn on more and the effect keeps happening until both transistors cannot turn on any more.
The value of the capacitor determines how long it takes for this action.
When both transistors are fully turned on, the capacitor keeps charging and the charge-current reduces. This causes the second transistor to turn off slightly and the two very rapidly completely turn off.
The charge on the capacitor is removed via the base of the second transistor and the resistance of the finger starts to charge the capacitor to repeat the cycle.
The resistance of the finger determines the time it takes to charge the capacitor and this creates the frequency of operation of the circuit.