>Is there a good resource to learn more about this without a Biology degree?
Immunology is
complicated. Some people call it the second most complicated system, behind the human brain.
Someone else has mentioned the Kurzgesagt resources, which are great. If you want something more formal, OME is also a great resource I think.
https://onlinemeded.org/spa/immunology
>I always assumed the body generated matching snippets at random and the law of large numbers would result in a match eventually.
Right. Antibodies have a Y shape, where the two tips of the Y are variable, and the base is fixed. The IgM are the completely random snippets that happen to match a wide variety of things. However, because it's chemistry, it's more subtle than match or no match. Think of it like a variable degree of stickiness.
When a pathogen is found, your body is going to iteratively mutate new antibodies and select those that bind the best to the collected antigens, until you have something very specific (= sticks very readily). I'm simplifying a little bit, but that's the process for generating IgGs that I was talking about.