Looking for intuition and/or insight regarding the "into-internalization principle" in category theory. Observation 0. Suppose $I$ is a set. Then the category $\mathbf{Set}^I$ can be explicitly described as the category whose objects are as follows: an object is a set $S$ equipped with a function $$I \leftarrow S.$$
Hence $\mathbf{Set}^I$ is (or at least, can be identified with) the category of all $I$-graded sets.
Observation 1. Similarly, suppose $M$ is a monoid, and $\mathbb{B} M$ denote the corresponding category with one object. Then $\mathbf{Set}^{\mathbb{B}M}$ can be explicitly described as the category whose objects are as follows: an object is a set $S$ equipped with function $$S \leftarrow M \times S$$ satisfying the axioms for a monoid action.
Hence $\mathbf{Set}^{\mathbb{B}M}$ is (or at least, can be identified with) the category of all set-theoretic $M$-modules.
"Definition." Lets call the phenomenon - whereby functors into a category (or category-like entity) can sometimes be understood "internal" to that category - the into-internalization principle.
Okay:


*

*I assume there's people out there who have a really good understanding of this principle, even if they've never bothered to give it a name. 

*I want to understand it better.


With the above goal in mind, I've compiled a short a list of questions.

Questions.
Q0. Does anyone know a slick proof of either/both of observations 0 or 1 that avoids concrete details and brings out the "true reason" why this works?
Q1. What are some other examples of the into-internalization principle?
Q2. What is the most general context in which this happens?
Q3. Are there any useful heuristics regarding when this will tend to happen?

I define an "answer" to this question as: any body of text that answers, or attempts to answer, one or more of the above questions. I've tagged with "soft-question" in acquiescence to the fact that giving a completely definitive answer probably cannot be done.
 A: I think you are pointing to two different phenomena, and I don't know which one you're more interested in. Can you clarify? 
In the second example you're just unwrapping the definition of a functor $BM \to \text{Set}$. You can always describe what a functor $C \to D$ looks like in a manner "internal to $D$" in a sort of trivial way: of course it's the same thing as a collection of objects in $D$ (one for each object in $C$) and a collection of morphisms between them (one for each morphism in $C$) satisfying etc. There's something interesting to say here but you'll have to be more specific about what, if anything, you think is going on here that needs further explaining. 
In the first example you're observing that $\text{Set}$ is the "classifying space for bundles of sets," in the sense that an $I$-indexed family of sets $X_i, i \in I$ is the same thing as a "bundle" $X \to I$ of sets (where $X_i$ corresponds to the fiber of this bundle over $i \in I$). There is a very general "yoga of classifying spaces" you can talk about here that goes in many different directions. For example, $\text{Cat}$ is the "classifying space for bundles of categories" in a way that is made precise by the Grothendieck construction. 
Edit: Okay, I think I know what interests you about the second example now, so just to check: is the point that the monoid $M$ is itself also a set? If so, the description you give generalizes to actions of a monoid $M$ in $\text{Set}$ on objects of any category $C$ which has all coproducts; such categories are tensored over $\text{Set}$, and this allows you to make sense of what a map $M \otimes c \to c$ means when $M$ is a monoid (in $\text{Set}$) and $c$ is an object; namely, $M \otimes c$ is just $\coprod_M c$. 
There is also a very general thing to say here involving monads and monadic adjunctions, but I'll wait for you to clarify your question before I say it. 
