I am a bit puzzled on how the colpitts oscillator part of this circuit works specifically the feedback with the emitter. Can someone explain please?

 

You chose a circuit that does different things with lower parts count which makes it hard to see the forest through the trees.
If you start out with a basic modulation this type of mixer the explanation can be moderately difficult chewing gum and concept multitasking at the same time for me it is better to combine typical models that have be scaled to have complimentary impedance values.
Because that is how I understand it when you combine transmitter and modulator and ask about Colpitts much could go wrong in explaining.
It can be done but it much easier to find a video on Colpitts then find a video on transmitter (a good video tutorial) is what works for you.
It becomes more clear in identifying the various sections: the mic amplifier, the bias resistor network, the resonant tank, the capacitive divider, the feedback path, the mixer that you can see in the schematic and output stage.