Two things, in particular, stand out for me when thinking about Haskell this way (as a "tool for thinking" language).
First, unless you're a mathematician, you probably haven't thought very deeply about algebraic data types, and how useful and expressive it is to build up a program representation from a collection of parameterized types. The article touches on this a little bit in noting that Haskell teaches you to think about data types first.
But it's more than just "data first," for me, at least. Grokking Haskell's type system changed how I think about object-oriented programming. Classes in, say, Java or C++ or Python are a sort of weak-sauce version of parameterized abstract types. It's kind of mind-blowing to make that connection and to see how much more to it there is.
Second, monads are a really, really powerful way of thinking about the general idea of control flow. Again, the most useful analogy might be to object-oriented programming. When you first learn to think with objects, you gain a flexible and useful way of thinking about encapsulation. When you learn to think with monads, you gain a flexible and useful way of thinking about execution sequencing: threads, coroutines, try/catch, generators, continuations -- the whole concurrency bestiary.
I think monads are hard for most of us to wrap our heads around because the languages we are accustomed to are so static in terms of their control flow models, and so similar. We're used to thinking about control flow in a very particular way, so popping up a meta-level feels crazy and confusing. But it's worth it.