Can you please explain the RC network and the dual pot? And how do you get the additional gain?

Yes.

As a pre-requisite you need to understand the RC phase-shift oscillator in its more common form (feedback from collector to RC stage). This infomation is available in the internet (just search by "RC phase-shift oscillator"). The following information presented here is top secret and will be buried in this thread and never found again

There are four different feedback topologies in amplifier design: series-shunt, shunt-series, shunt-shunt, series-series (use internet search if you want to know more). The "normal" RC phase-shift oscillator uses the shunt-shunt feedback model. The emitter-follower circuit (with only bias and emitter resistor) is already a series-series feedback amplifier on its own, where the FEEDBACK voltage is 180 degrees out of phase from the OUTPUT. See the diagram below:

In this image, we have the signal source added and the feedback voltage Vf is summed as "inverted" to the input. Please ignore the RC/RL, since this model can be applied to the common-emitter amp as well but not now. Hopefully this is clear so far. In the oscillator, the signal source can be removed, since the input signal is taken 100% as a feedback signal from the output.

Next we draw the Uni-Vibe RC network on top of the same feedback diagram (NOTE: redrawing circuits in standard form is a huge help to understand them)

Now it is starting to look as the "normal" RC phase-shift oscillator. So we have 180 degree shift already at the input side of RE and another 180 shift from the RC network. This sums up either as 180 - 180 = 0 OR 180 + 180 = 360 = 0 at the frequency of oscillation. When there is positive feedback (0 degree shift) the thing will oscillate. But wait...

There is also a theory called the "Barkhausen stability criterion". It says that for oscillation to happen, the LOOP GAIN of the circuit must be equal or larger than -1. There is so much false information floating around saying that the gain of the circuit must be so-and-so much for oscillations to happen (to make up the loss of the RC feedback network). This is not so. The basic emitter follower circuit can easily have a LOOP GAIN of over 100 (although the voltage gain is ALWAYS less than 1), and there is no magic related to this. The loop gain is calculated differently than "normal" gain. So in this case the Barkhousen stability criterion is fullfilled and also overly ensured by the darlington's high current gain (these days a single high-Beta tranny like BC549C would be enough here).

By adjusting the resistances, you adjust the cut-off frequencies of the RC-pairs, thereby affecting the oscillation frequency. Not any different than in the normal RC phase-shift oscillator.

And if your next question is how to calculate the loop gain for this specific circuit, then I am sorry to say but you are not ready for it. I did it using numerical math tools and utilizing matrix algebra. There is no simple equation to give.