How are API changes announced? Are emails sent out to developers, or is there someplace I can subscribe to?
They're not announced via a dedicated channel, perhaps because there haven't been too much changes in years. Version 2.2 has been the latest stable release and while there is a newer 2.3 version to support the official mobile apps, it hasn't been released and probably never will be since the apps are deprecated.
We regularly discuss the API on Meta Stack Exchange as an often-forgotten part when new changes are being rolled out (e.g. the new close reasons or the renaming of favorites to bookmarks) but those are relatively minor changes (sometimes the format doesn't even change). And there's no good way to monitor those changes either.
A year later, 2.3 has been released and the company chose an announcement on Meta Stack Exchange as medium.