Monads are popular for side effects because they have an implicit notion of sequencing, so evaluating a monadic expression enforces the sequence of operations. Works out nice for IO and Futures and so on. But List is also a monad, and flatMap (as many other languages call it) doesn't inherently have any side effects at all. Same goes for Maybe/Option (essentially a list of zero or one element), and State (which does take advantage of sequencing).

It's all about getting an intuition for how many things fit the shape:

    flatMap :: m a -> (a -> m b) -> m b

Where "flatMap" might have different names in different types. Once you see that pattern in some code, you'll start seeing it in a lot of other places.

Most monadic effects aren't executing side effects.

Nope. For example lists are monads, but doesn’t have anything to do with side-effects.

Monads are just a pattern for chaining functions.