This a tuned, relaxation oscillator called Royer, or Baxandall. Variations of the configuration are commonly used in DC to AC converters and inductive heating transmitters.
An important characteristic of this type of circuit is the tuning of the primary coils. In this case, the tuning capacitance is the series combination of 4u7, 100n and 4u7.
In many Royer-type oscillators the feedback to the base of the BJTs (or to the gates of FETs) is provided by an extra winding in the transformer. In this ingenious circuit, the feedback is by means of a capacitance divider (a bit like the resistor equivalent) so an extra winding is not needed.
The two 22 Ohm resistors limit the base current at switching.
There is no protection against reverse Vbe on the BJTs. However, the value of primary inductance is needed in order to simulate how close the BJTs are to destruction!!
Answers to 3 questions in comments
Would the transformer primary inductance be part of the tuning
requirements to make this oscillator work?
The usual method of oscillation makes use of transformer saturation. At saturation, the collapsing magnetic field in one primary winding triggers positive feedback to switch the current to the other primary winding. Thus, the primary might not have a constant inductance during operation. EL wires are operated at a frequency of about 1 kHz, I think.
The transformer steps from 3V to 120V. Is this due to the resonance
rather than turns ratio?
Voltage step up is due to turns ratio only.
120 VAC is high enough to be lethal - be VERY careful.
The resonance helps with power efficiency: there are smoother signals and fewer wasteful high frequencies.
The approximately 1 kHz frequency
i) allows you to make a smaller transformer (than, say, for 50 or 60 Hz); ii) eliminates flicker of the light.
Would an improvement be needed or anything added to this circuit for
the issue of no reverse Vbe protection for safety?
"Safety" around Vbe is only for the transistor, but see the previous note about 120 VAC.
This type of low cost circuit is finely balanced in terms of good performance from a few components.
If anything is changed the balance will be upset.
The good news is that the supply voltage is low and (I think) the current on the secondary side will be low - so, overall, low power. Hopefully, the BJTs will survive.
If your BJTs fail frequently, excess reverse Vbe might be the reason.
Try: either diodes in series with the bases, or, diodes anti-parallel with the bases.
You might have to make other adjustments to make it work again...
This is tricky analogue circuit, Good Luck!