I am now trying to understand better what "abstract interpretation" in programming languages are. I found a good book chapter that explains the idea of extending the domain with a least fixed element, the four axioms that yield a fixed point for a continuous function, and so on. I understand these technical details (though I am not quite sure what is "abstract interpretation" exactly referring to in this whole scheme).

What I am not sure about is what motivates the use of abstract interpretation? Is it just identifying fixed points for computable functions? Is the main motivation coming from having recursion in most programming languages?

Would also be glad to get some high-level overview which does go technically deep enough for someone who has a degree in computer science. I find the Wikipedia page rather unsettling.