Axiomatic Metacategories In Mac Lane's Categories for the Working Mathematician, he starts off with the notion of a metacategory, where we have a "collection" of objects a, b, c...; a collection of arrows f, g, h..., and then some extra structure (axioms ensuring things like cod $f$, id$_a$, etc., are well-defined). He then says that a category is "any interpretation of these axioms within set theory." 

It seems we have this so we can talk about things like a "metacategory of sets." But I'm failing to grasp how this definition is strictly axiomatic, or how it escapes from set theory. In other words, what is a collection, if it isn't a set? 

Edit: In my effort to figure this out, I came across an explanation that metacategories are models of a particular first-order theory, and categories are a subset of those which conform to set theory. This makes sense, but I'm fuzzy on this area of math and its details. In particular, how does a presentation in terms of symbols and sentences defend a presentation which talks about actual things, like collections and functions?
What MacLane is doing is basically providing a definition of what a category is in any theory of collections.

Here he uses the term collection in a very technical sense as oppesed to set.

Without going to much in the details you can think to collections as to thing for which it makes sense to say that they have elements. Sets are collections which are elements of other collections, these are the collections of theories like ZFC.

As you pointed out this is required to build things like the category of sets. The problem is that if you had only collections that are sets then you wouldn't be able to provide a collection of all sets (which cannot be a set by Russell's paradox), hence you wouldn't be able to provide a category of categories.

If you admit collections that aren't sets (as for instance in NBG) you become able to build a collection of all set (which will be a collection, though not a set).

Hope this helps.
Unfortunately, you are in an area where people contemptuously defend textbook summaries against what is found in original sources with insulting language involving "history" or "philosophy".

What you need to look at is the distinction between intension and extension as the foundations of mathematics evolved in the late nineteenth century and early twentieth century.  I have not personally found a convincing defense of category theory in terms of extensionality.  So, MacLane's discussion of metacategory should be understood in the context of intensions.  The import of extensionality in the foundations of mathematics can be understood by critically considering Frege's distinction between "a concept" and "the extension of a concept".  Concepts are intensions.  The extension of a concept corresponds with the collection of individuals with respect to which a concept can be instantiated as being true.  And, one ought not overlook the significant role of the definite article "the" in the expression "the extension of a concept".  

In Frege's writings there is an intense criticism of "intensional logicians" because intensions are not grounded by a semantic conception of truth.  Today, this difference would be characterized by the contrast between a correspondence theory of truth and a coherence theory of truth.  To say that mathematics is extensional is to say that its semantic theory is based upon a correspondence theory.

This is why one speaks of Tarski's semantic theory as "materially adequate" or the classical connective for implication as the "material conditional".  Tarski's theory treats terms as denoting objects, and, truth tables as functions have a domain over the "truth objects" introduced by Frege.  What is "material" here corresponds to the fact that what is presumably denoted are extant individuals.

Generally speaking, functions and collections are not viewed as primitive from an extensional point of view (with regard to "truth") because they are not individuals.  This originates with Aristotle who called individuals "primary substance" and the collections of individuals making up "classes" as "secondary substance".  The Aristotelian system did not treat of collections as objects.  So, one must make an adjustment for what happened when Russell introduced his theory of types.  Individuals continue to ground the theory, but, instead of just one notion of "secondary substance" related to partitions of a universe of objects, one must invoke a hierarchy of types based upon the predicate "is a set of".

The distinction between intension and extension motivating Frege's work arises with Leibniz.  He describes his own view of logic as inverting the order relation of the Aristotelian term logic semantics.  This is the origin of the identity of indiscernibles which has been rejected as a logical principle by the modern philosophy community (in spite of generating contradictions when combined with the substitutivity interpretation of identity in the first-order calclus).  The Aristotelian logic is extensionally grounded.  The Leibnizian logic is intensional.

With regard to your question involving the difference between collections and sets, it is becoming fashionable to invoke mereology as a notion of "aggregate" that may not be the same as a collection.  Michael Potter maintains that the conflation between "membership" and "subset" is a certain sign of mereology and claims that one can find it in Dedekind's work.  On such an account, some of the business of the last 150 years has been to clarify the distinction between "membership" and "subset".  Arguably, the many valued properties of first-order logic (obtaining over individuals comprising set domains) is the result of this hard work.

One of the problems with thinking about "sets" in terms of "collections" is that there are many in the philosophy community who think about "sets" in terms of "extensions" (by which I refer to comprehension axioms).  In my non-standard and uninformed view, Cantor concerned himself with collections and the arithmetization of mathematics.  That is a far cry from extensions of concepts given through comprehension.

I cannot reconcile the arguments of others.  What I can tell you is that if you view sets as collections, the advocates of first-order logic as mathematical logic will reject many conclusions because  collections are  fundamentally intensional and second-order logic does not have the nice properties of first-order logic.  What one has are two views of set theory which are not clearly delineated in the introductory literature.  One has the first-order interpretation corresponding with Skolem's criticisms of Zermelo.  Then one has the iterative conception of set which corresponds more closely with a view presented by Zermelo after those criticisms.

And then, of course, there is the category-theoretic work of Lawvere who attempts to justify his positions on the basis that Cantor's views had never been properly represented in Zermelo's axiomatization.  Since Cantor developed his ideas as a contemporary of Dedekind, a number of his positions may also be tainted with the mereological conflation discussed by Potter.  So, Lawvere's justification may have support.  But, it will very likely lie with intensional mathematics.

I wish you all the best with finding an answer to your question.  I certainly have not yet answered it for myself.  But, before you let anyone convince you of a first-order theory of metacategories, let me suggest that you read the paper on natural equivalences by Eilenberg and MacLane that led to category theory.  It is far better to trust yourself reading original sources than it is to trust textbook authors.  And, you should try to understand what is being represented before accepting that a purported representation is faithful simply because it is "symbolic".