It is supposed to be a simple inverter, but it has me completely confused. How is this circuit supposed to work? How does it oscillate?

 

It doesn't, the drawing is wrong the transformer goes to the collectors, and each base resistor goes to the opposite transistor collector. circuit

 

The drawing is half-wrong. The upper transistor is ok. The lower transistor collector and emitter are reversed.

Note that your circuit is more of a concept than a working schematic, is not very efficient, and the frequency of the AC output varies significantly with temperature, load current, and from one build assembly to the next.

ak