What is this "System software from developers" security feature called and where can I find the technical details for how it works with applications (like AVG)? I believe this is a kernel extension, but how exactly does it work?
It is called Gatekeeper on Mac.
How it works.
It verifies at bare minimum if a app is:
blacklisted, code-signed, code-signed by Apple or a certified developer, code-signed and the code-signed contents still match the signature.
As you can see from your screen you have control over it.
As of High Sierra, Apple has removed the Option to Download from Anywhere, but you can still enable it if you whish, or use Allow as in your screen.
AVG Technologies CZ, s.r.o. is a Czech security software company headquartered in Brno, that was founded in 1991 by Jan Gritzbach and Tomáš Hofer and has been a wholly owned subsidiary of Avast since 2016.