Need help on 2N3055 12v DC to 220v AC Inverter Circuit
This kind of oscillator depends on the transformer reaching saturation. Most simulators only have "ideal" transformers, and will not show the behavior required for oscillation.

If it weren't for saturation, one transistor would switch on and the current through it would increase indefinitely. In a real transformer, the core reaches saturation and this causes the base drive for that transistor to disappear, triggering the circuit to switch to the opposite state.