Why is this? What relation to the gods have to anticipations?
You'll need to read at least the DeWitt material before you get a more informed view on this. As I recall there's not much on it in Emily Austin.
As to how it relates to anticipations, that's part of Epicurus' theory of knowledge that responds to Plato's ridiculous theory of forms and pre-existence of the soul and the impossibility of knowledge without both of those working together. Plato held only the eternal is real and only the ideas/forms are eternal. Thus nothing revealed by the senses and rational processing of the sensory data is truly real.
Epicurus rejects that totally and says there are three faculties of contact with the real world - the 5 senses, pain and pleasure, and anticipations. Anything they present to you is "real" and but it's up to you to understand in what way they impact you and react accordingly. Epicurus concluded that in putting all the evidence together it's obvious that nature never makes a single thing of a kind, that space is infinite and eternal, and that it is simply logical to believe that earth and humanity are not the only location of life in the universe, or the highest life. Some forms of life are more successful than we are at living happily, even to the point of conquering death -- but nothing and no one is SUPERNATURAL.
If you start saying "there's no direct evidence of life anywhere else that I've ever seen" then you're eliminating many logical conclusions that flow from the evidence we do have. It's not necessary for us to have seen or touched everything to believe that it exists -- we believe atoms exist without ever seeing or touching them.
So there are many deep implications of the physics and epistemology that you never get to until you get past the superficial idea that Epicurus only cared about "pleasure" and nothing else.
So perhaps it’s best for me to drop my modern understanding of the term “gods” when I read what Epicurus and his fellow philosophers have to say on this point.
That's absolutely essential and you'll see that as you read more. To repeat, that's the danger of focusing only on the ethics -- you miss the major part of the philosophy, including the parts that establish the ethics in the first place.
The letter to Menoeceus is only a part of the philosophy. It's presented much more fully in Lucretius, but most people aren't going to be able to pick up Lucretius and get the full point because it's so unfamiliar to modern ways of thinking. That is why I suggest reading the DeWitt book very early in the process of studying Epicurus so you can begin to see the outlines of the full picture as quickly as possible. It's far from perfect but it doesn't focus exclusively on the ethics and dismiss the rest of the philosophy like many other approaches do.