Explain this circuit please.

[tarakan]

How to get more voltage from a fixed output transformer

What would happen if this transformer had capacitors connected across it.
What would happen if the circuit had spark gaps and all the other attributes of your typical "free energy" device?

What is the official physics explanation of how this circuit works?
Is it related to the bifilar coil?

[bistander]

What is the official physics explanation of how this circuit works?

It functions as an autotransformer. Search for autotransformer to find the classic theory.