Simple self oscillating flyback converter
Your circuit looks like this:



simulate this circuit &ndash; Schematic created using CircuitLab
In reality, \$R_\text{S}\$ is probably not needed. But it is another degree of freedom, so it's not necessarily excluded. But for a simple circuit I'd probably eliminate it.
There is a problem with all these circuits. That is the fact that \$Q_1\$ above might every well have its \$V_\text{BE}\$ reversed-biased by far more than its avalanche voltage. So a diode may be helpful in this circuit to snip that.
If you carefully read my discussion here, you'll see why the basic circuit should work. The only addition here is \$L_3\$, which is arranged so that when \$L_2\$'s magnetic field is collapsing, \$L_3\$ is polarized correctly for \$D_1\$ as shown above.
So the stored energy is positioned well to be discharged via a drop across \$D_1\$ to charge \$C_1\$, over time.
It's not all that complex.
Please add questions and I'll try and answer them. But please also first read this discussion, too. I think it will make a lot of sense.