Oscillator circuit

It's a phase shift oscillator.

Normally, feedback from the collector to the base acts "negatively" and this is quite important for some amplifiers. This is because the collector signal is the inverse of the base signal (also known as 180º out of phase). Anything fed back does so without causing oscillations. This type of feedback is also used in op-amps for controlling gain.

On the circuit in the question there are a bunch of components that take the collector signal and phase shift it enough so that at a particular frequency, it appears in phase with the base signal and reinforces it. This makes it oscillate.

On a more technical level, the feedback formed around R2, R3, R4, C1, C2 and C3 act as a "mild" notch filter. It should be said that the intent of a "good" notch filter is to totally remove one frequency (such as 50Hz or 60Hz when mains AC is a problem). The frequency which is notched out will be phase shifted by 180º and if it isn't totally notched-out (as in a good notch filter) what remains will feed back and reinforce the original base signal causing it to oscillate.

It doesn't matter that the signal might be attenuated by 20dB, there will still be enough signal left to be amplified and generate a sinewave.

If you alter your diagram to look more like this it should work:

In your diagram, instead of a voltage divider and a capacitor you only have a single resistor. This won't work because the output of the Colpitts oscillator dips below 0.6 volts, and anything less than ~0.6V won't be picked up by an NPN transistor.

In the diagram, you can see that I've coupled the output of the oscillator with C2 to a voltage divider, with R1 and R2. The voltage divider adds a DC offset of 9V to the base of Q2. This way, the voltage at the base now ranges from 3V to 15 volts instead of -6V to 6V. Since the signal is always above 0.6V, Q2 passes it.

The output uses C1 to decouple the DC from the sine wave, leaving you with your original 6V amplitude sine wave, just with amplified current.

You can change the values of some of the components around if you want, but C2 should optimally be around 1/(2*pi*500KHz*R2), and C1 around 1/(2*pi*500KHz*R3).