How to build an FM transmitter and how does it work?
How am I supposed to learn this stuff?


There are no short cuts and I would learn the following (but there's no guarantee I haven't missed anything from the list because I learnt this stuff decades ago and I don't have perfect memory): -


Learn ohms law
Learn capacitors and inductors
Learn tuned circuits
Learn BJTs then how they are configured such as common emitter, common base, and common collector
Understand small signal AC analysis and DC analysis of these circuits.
Learn about active filters using BJTs (learning for op-amps will also be useful)
Learn a little bit about power supplies
Learn about modulation techniques such as AM and FM


When you are reasonably confident with the above you should be able to see that the FM transmitter in your question is a common base configuration at the carrier frequency and for the modulation signal it is a common emitter amplifier. Along the way to learning about BJTs you will have learnt about miller capacitance and if you did your learning correctly you'd have known that a signal on the base of a common emitter amplifier can "modulate" the miller capacitance.

Put all this together and you have your answer - the miller capacitance is modulated by the modulating signal on the base. This in turn modulates the resonant frequency of the tuned circuit in the collector and, feedback from collector to emitter creates an oscillator whose centre frequency is wobbled by the base signal.

Now don't expect anything more from me - go and do some learning and maybe a couple of months of intensive slog will get a keen beginner about 50% there.

Bear in mind that I have no idea what your starting point is so don't gripe at me if you already know about some of the stuff I've mentioned.