why am i able to wire one up to a 12vcar battery and magically get 120v ac?
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There are many ways it can be done. The simplest way is likely by using a transformer.
You can change the AC voltage level with a transformer. The difference in input vest output voltage depends on the number of winding in the input relative to the output side. This is the main reason AC is used in the power grid. It is simple to change the voltage, a higher voltage is more efficient in transporting the same amount of power than a lower voltage. At the same voltage, DC is more efficient than AC in power distribution the problem is it is a lot harder to change the voltage.
Transformers require AC, they do not work with DC. So how do you use one with a battery? The simple way is to just change how the battery is connected to the coil. If you flip the position of the wires to the battery and do that over and over you create AC. It is simple to do with a switch with multiple poles and then to physically reconnect the wires you can then use a specially controlled switch ie a relay. That is how it works in the past today transistors are used as switches to create AC from DC.
A DC motor do exactly this with a commutator, which is a rotation mechanical part that changes the direction in the DC is connected to the could in the motor. Conductive brushes typically made of carbon are pushed agas a rotational commutator with multiple separate conductive areas. Inverter does not use rotational parts like that but is quite simple way to understand how you can make AC from DC mechanically https://www.cs.uaf.edu/2015/fall/cs480/lecture/09_11_motor_control.html
Just changing how the DC is connected produces an AC square wave, not a sine wave like we have in the power grid. It is good enough for some application but not for all and more complex designs exist that create sine waves
https://en.wikipedia.org/wiki/Power_inverter
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