How does this power inverter circuit work?
The simplified diagrams in that article are only meant to be suggestive of the general circuit topology, not taken as complete working circuits.

This particular power oscillator won't reliably start up on its own without additional components. But if you assume that one transistor conducts more than the other, this induces current into the center-tapped primary winding. The two feedback windings are connected such that the drive of that transistor is increased while the other is decreased. In short order, the conducting transistor is saturated and the other is cut off.

The primary current continues to increase until the transformer core saturates, at which point, the voltages from the feedback windings start to switch polarity. This causes the conducting transistor to cut off and the other transistor to switch on. Now the primary current builds up in the other direction until the core saturates in that direction, at which point, the whole cycle starts over again.

The oscillation frequency depends primarily on the effective inductance of the transformer's primary winding, the input voltage, and the saturation current of the transformer. Most importantly, it varies significantly with load, which makes the circuit completely inappropriate for any load that requires a particular frequency.