Suppose $C=(C_0,C_1,d_0,d_1)$ is an internal category in the category $E.$ Then, following Maclane's definition, a (left) object over $C_0$, is an object $\pi:H\to C_0,\ $ together with an action map $\mu :C_1\times _{C_0}H\to H$ that satisfies the usual unit and associative laws.
Maclane's example for $E=Set$, and $H:C\to Set$ takes the object function $H_0$ and defines $\pi:\coprod _{c\in C_0}H_0c\to C_0$ and then the action map $\mu$ I assume would be, for $f:c\to c',\ x\in H_0c,\ \mu(f,x)=H_0f(x).\ $ I have checked that the pertinent diagrams commute so $\pi$ is indeed a left- $C$ object.
My question has to do with Maclane's comment to conclude the section that this construction includes the functors $H:C\to Set$ when $E=Set.$ What exactly does this mean, since one already $starts$ with the set valued functor $H$ and then proceeds to construct the $C-$object. Is this construction intended to remedy the fact that $Set$ itself does not correspond to an internal category? Or does it give an extension in some way of an internal functor to the ambient space? More generally, what are some applications of this idea?
I think there is a sense in which you're right, that the construction is, spiritually, how you remedy the fact that $\mathbf{Set}$ isn't a small category. That is, once you have the idea of an internal category and an internal functor, which over $\mathbf{Set}$ gives us small categories and their functors, that still doesn't allow us to formulate presheaves internal to $\mathbf{Set}$, because the codomain of such a functor simply isn't (and, generally, can't be) a small category. Enter our (left) $C$-objects, which, as noted by others above, make sense in any category with enough structure and, in the case of $\mathbf{Set}$, correspond precisely to $\mathbf{Set}$-valued functors.
As for applications,
It's of course interesting any time you're trying to look at a category as a metamathematical setting, to see how category theory runs.
It also gives you notions of internal (co-)completeness, though it doesn't take much for a category to be internally (co-)complete.
In topos theory, the collection of internal diagrams on $C$ (a synonym for $C$-objects) is also a topos, providing extra places to look for (counter)examples.
Occasionally, some interesting subcategory of a category $E$ can be "packaged" as internal categories (that is, as full internal subcategories), and one can either simplify reasoning about the subcategory by dealing with the internal category, or produce interesting counterexamples regarding internal categories from the properties of the subcategory.
IMO, the most fun feature of them is how they generalize; because the construction Mac Lane gives is actually just the special case of a construction that can be applied generally to fibrations over finitely complete categories. Without going into too much detail*, the $C$-objects in $E$ correspond to special functors from a category constructed around $C$, sometimes written $\mathrm{Fam}(C)$, to the arrow category of $E$. Given an appropriate functor $f:D\to E$ (one that makes a fibration), the right kind of functors from $\mathrm{Fam}(C)\to D$ will essentially be $D$-valued diagrams on $E$-internal categories. I can't think of a great number of areas where this is "useful," but it's cool in any case.
* I have written this assuming that you don't know anything about fibrations and might just be encountering internal category theory for the first time. If this assumption is wrong and the result annoying, let me know and I'll fix it.