This is genius. As someone who is familiar-enough with minor electronics to fuck something up, but not confident enough to look at this photo and go for it—what am I trying to learn here? What are the terms I'm trying to google to figure out how to connect <electronic board> to <electronic board> via <wires>?

One of the articles on this mentioned "ratgdo" as a simple board to do most of the "make a button wirelessly available to homeassistant", I haven't tried it but searching on it gets you a lot of reasonably specific articles and videos.

The pictured solution is very easy and comes with directions (ratgdo).

However, if even that is too much you can make a Switchbot do almost anything. It's just an actuator that pokes buttons and is a premade product with a shell rather than a DIY thingy.