It may be more than 250 amps and it could be a lot less depending on how much load power you are taking. If the inverter is rated at 3 kW this will be the maximum output power it can deliver.
Given that an inverter might only be 90% efficient, the input power could be as high as 3.333 kW and then the current from a 12 volt battery would be 278 amps. Of course, the inverter may have a surge power rating of 4 kW and then the surge current taken from the 12 volt battery might be as high as 370 amps.
Modern inverters can both provide and absorb reactive power to help grids balance this important resource. In addition, because reactive power is difficult to transport long distances, distributed energy resources like rooftop solar are especially useful sources of reactive power.
Inverters are just one example of a class of devices called power electronics that regulate the flow of electrical power. Fundamentally, an inverter accomplishes the DC-to-AC conversion by switching the direction of a DC input back and forth very rapidly. As a result, a DC input becomes an AC output.
In order to provide grid services, inverters need to have sources of power that they can control. This could be either generation, such as a solar panel that is currently producing electricity, or storage, like a battery system that can be used to provide power that was previously stored.
Fundamentally, an inverter accomplishes the DC-to-AC conversion by switching the direction of a DC input back and forth very rapidly. As a result, a DC input becomes an AC output. In addition, filters and other electronics can be used to produce a voltage that varies as a clean, repeating sine wave that can be injected into the power grid.