How does the signal actually change from DC to AC within an astable multivibrator inverter?

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I’m going to paste the circuit from the linked article here, so it’s easier to discuss:

The ‘astable multivibrator’ part is everything in the circuit other than the two IRF640s and the transformer.

The name ‘multivibrator’ is a rather quaint description of a simple RC relaxation oscillator. It’s called that because it produces square waves, which contain a great deal of harmonics or overtones of the base switching frequency; that is multiple vibrations. The linked article erroneously claims this circuit produces “a closely resembling sinusoidal waveform” [sic], which is simply untrue.

To understand its operation, it’s best to look at the circuit after it has been running for a while, because the situation at the moment that power is applied is a bit hit-and-miss. So let’s assume the the left-hand 2N2222 (I’ll call this TR1) is switched ON, and the right hand 2N2222 (TR2) is OFF. For this to be true, the base of TR1 must be above 0.6v, and it gets its base current through the 16.5K resistor. When TR1 is ON, its collector terminal is close to 0v. On the other side, TR2 is OFF, so its collector terminal is at +12v. The left-hand capacitor will now start to charge up through the 16.5K resistor until the voltage reaches 0.6v. This switches TR2 ON, bringing its collector to 0v. This brings the other side of the right-hand capacitor sharply negative, which instantly shuts off TR1. Now the right-hand capacitor starts to charge up from the negative voltage to +0.6v, and once it reaches that value, TR1 turns ON again, bringing its collector to 0v, which causes a sharp negative spike on the base of TR2 through the left-hand capacitor. This shuts off TR2, and the cycle repeats, endlessly. (Note that the diagram shows the capacitors as electrolytic types, but since they are operating in both directions here, you should never use electrolytics in such a circuit.)

So anyway, we now have the two 2N2222 transistors switching ON, and when one does, it switches the other OFF, and then after a short delay while the capacitor recharges, the opposite happens, in an endless cycle. Tick-tock, tick-tock. The two IRF640s are power MOSFETS which switch ON when the gate terminal is at a high voltage compared to the source (S) terminal. So these also switch in antiphase. They are connected to the primary of the transformer, and the centre-tap is fed with +12v. Therefore, when one MOSFET is on, current flows through half of the transformer primary winding in one direction, then when the other switches on, it flows the opposite way through the other half of the transformer primary. The voltage is stepped up by the transformer which also eliminates some of the higher harmonics in the switching, so what you get at the output is a very crude AC supply with a lot of remaining harmonics. As an inverter, this is pretty poor, and will probably not work very well powering anything sensitive. A good AC signal is a pure sine wave, but this circuit is not going to produce that.

The frequency of the switching is set by the 16.5k resistors and the 1µF capacitors, and is ~0.693 x 2RC, (the linked article quotes an incorrect ‘formula’ for the frequency) which in this case is about 44 Hz — near to, but not very precisely, the mains frequency — and in any case, component tolerances will affect this, especially capacitors, which are not usually precision components. I wouldn’t recommend this as a way to build a real inverter — the output is too dirty and the frequency isn’t well controlled. I would also caution that the output from this could still be just as lethal as ‘real’ mains voltage, so if you plan to build it, be careful.