Sorry if I was a bit snarky in my first reply. As far as I understand this, you can't explain the structure of the Ulam spiral in a trivial way. You are correct that there is a trivial part to the spiral: diagonal lines alternate between odd and even numbers, therefore all the primes lie along diagonal lines. However, the structure is much richer than that.

The question is why certain lines have lots of primes and not others? why is a given polynomial so rich in primes, while other similar ones are not? That's what some of these conjectures are trying to prove.