How does a DC to AC inverter work?

Start with a switching circuit that supplies the DC with one polarity then reverses the polarity, back and forth. That’s done at the target AC frequency. An inverter that’s any good will knock the corners off the square wave and will further attempt to smooth the wave to more closely resemble normal AC power from the mains.

This is why many devices will not work on inexpensive inverters. The quick and dirty way is to make a modified sine wave by what is referenced above. A proper inverter will add circuitry to make a true sine wave