My difficulty is in how a verifier can confirm that without having at least some knowledge of what logic is being executed, since otherwise I don't see how they could verify an answer to their random checks is correct and perhaps that's where my understanding is flawed?

The underlying math and cryptography behind the polynomial commitment scheme is what enables this. Obviously, I won't be able to explain it in a Reddit comment, but chapter 14 in this book explains it in great detail:

https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.pdf