Natural occurrences of monads that make use of the category-theoretical framework
Asking whether an occurrence of monad is natural is similar to asking whether a group (in the sense of group theory) is natural. Once you formalise something, in this case as an endofunctor, either it satisfies the axioms of being a monad or not. If it does satisfy the  axioms, then one gets a lot of technical machinery for free.

Moggi's paper Notions of computation and monads pretty much seals the deal: monads are incredibly natural and useful for describing computational effects in a unified way. Wadler and others translated these notions to deal with computational effects in functional programming languages, by using the connection that a functor is a data type constructor. This  adds the icing on the cake. FP monads allow a treatment of computational effects such as IO which would be extremely unnatural without them. Monads have inspired related useful notion such as arrows and idioms which are also very useful for structuring functional programs. See the Wadler link for references. FP monads are the same thing as category theory monads in the sense that for an FP monad to work the same equations need to hold --- the compiler relies on these. Generally, the presentation of the monad differs (different but equivalent operations and equations), but this is a superficial difference.

A vast amount of the work of Bart Jacobs, to take but one example, uses monads. A lot of work stems from coalgebra, which is a general theory of systems. One of Jacobs' (many) contributions to the area is the development of a generic notion of trace semantics for systems (described as coalgebras) based on monads. One could argue that the notion of trace semantics is natural: What is the semantics of a system? The list of actions that can be observed!

One way to understand monads is to first program in Haskell using monads. Then find one of the many good tutorials available (via google). Start from the programming angle, then move to the theoretical side, starting with some basic category theory.