How does an audio door intercom powered by solar operate?

It operates exactly the same way as any other intercom. The only difference is how it gets its power. As an example, the intercom at the following site can run on battery power, AC power, or it has a solar power option.

Whatever intercom you use, you'll have to consider that an intercom near a doorway may be shielded from the sun so the solar panel will likely have to have a long cord on it.

You may be able to find some sort of solar panel you can use to convert any battery-powered intercom to solar. Just look for one that has the right voltage and current output to match the intercom.