As a beginner, I can assure you that without someone explaining the point of the arrows, this is most certainly not for beginners.
I'm no expert, but from my understanding, there is nothing more to be said (other than stating the composition / identity laws): you're at a level of abstraction where you cannot say anything more. Objects. Arrows. An arrow goes from an object to another.
If it's any help, you can think of them as mathematical functions, where the origin point is the function's domain and its destination point the function's range (called co-domain in category theory, because why not).
Bear in mind that while this example is correct, that's all it is: a concrete example of an abstract notion.
If it feels simplistic, that's because it is. Arrows are a trivial notion. It gets more complicated when you study different kinds of arrows (homomorphism, isomorphism, whateverism), or arrows in "concrete" categories (functors are arrows in the category of categories, natural transformations are arrows between functors...).
In the book "Conceptual Mathematics" the concrete example is finite sets and mapping between finite sets. That is way more understandable than 'objects' and 'how to go from one object to another'. It's just too abstract for a general audience I believe.
Conceptual Mathematics needs concrete example to explain the algebra of function composition, and more importantly to prove certain properties and theorems. This presentation's goals are far more modest and the abstract approach allows it to not take up 700 slides.
That being said, I don't think I'd have been able to go through this presentation in its entirety if I hadn't had some previous exposure to the field. So while I disagree with you for discussion's sake, I secretly think you're right.
The point of these instructional methods is to build an 'intuitive sense' or "system 1" knowledge of category theory. Having a lot of easy to grasp examples to help you start forming the theory in your head i find quite valuable.
They start off with objects, what are they? Are they sets? I'm going to assume they're sets. Then the arrows represent mappings, something like a function that takes an input and maps to stuff in one of those other objects (set of objects?). Then the codomain is the set of objects you can get to by calling that function on the domain of objects.
Am I in the right ball-park here?
Treat objects and arrows abstractly like you're building a graph. Actual meaning can be applied later. A category is thus a set (or proper class) of objects, O, and a set (or proper class) of arrows, A along with two mappings dom and cod from A -> O. The laws are first that for any object (o in O) there exists an arrow (id(o) in A) such that dom(id(o)) = o and cod(id(o)) = o and second that if you have two arrows (f, g in A) such that cod(f) = dom(g) then there is defined their composition (compose(f,g) in A) such that dom(compose(f,g)) = dom(f) and cod(compose(f,g)) = cod(g).
You can draw all that as the objects and arrows and things these slides display. So far this just gives you a graph-like structure.
The "category laws" ensure coherence of the category and state that (compose(f, id(x)) = f) when defined, (compose(id(x), g) = g) when defined, and that compose is associative. From an abstract algebraic POV this is almost like saying that A is a monoid which ought to (correctly) suggest that most of the interesting structure of a category is carried by the arrows.
Now, if you want to talk about specific categories you can do what you were doing. For instance, the category of sets, Set, is the category where you take O to be the proper class of all sets and A to be the functions between them. Immediately, you'll find that dom(.)/cod(.) are the domain and codomain of your functions, id(.) locates the identity function on a set, and compose(.,.) composes functions.
As a more interesting example you could consider Grp the category where O is all groups and A is all group homomorphisms. This is a subcategory of Set obviously and it also comes with much more interesting structure. For instance, an important category theoretic statement about Grp is that Grp has a "zero object", which is to say that (a) there exists an object init such that there is a unique arrow from init to any other object, (b) there exists an object fin such that there is a unique arrow from any other object to fin, and (c) init = fin.
The objects here are not necessarily sets, and the arrows are not necessarily functions -- this is actually one of the points of category theory, to abstract the notion of morphism away from set theory. This becomes necessary, when in your regular mathematical life you encounter morphisms which are not strictly functions, but nevertheless in many ways behave as such. Best example would be the homotopy classes of maps between topological spaces. Homotopy classes do not map points from domain space to codomain space, but nevertheless you can compose them, there's a homotopy class of identity, which acts as an identity on composition, sometimes there are homotopy classes that are inverses and so on.
If this sounds complicated and unnatural, that's because it is: it is pointless to learn category theory before learning where and how it is useful. This is completely opposite order than in which it was conceived, and totally misses the purpose the category theory, which is to make everything clearer, to reduce the number of notions, to make it easier to notice analogous behavior between separate concepts, to reduce number of repeating arguments. Learning what a morphism or functor or natural transformation is, without having tens of examples of these notions, only increases the number of concepts without a corresponding increase in understanding.
My advice is, if you want to learn category theory, learn algebraic topology first, or at least abstract algebra (groups, rings, modules, fields). Otherwise, you'll see very little purpose, and very little use.
While not strictly necessary, learning a bit of category theory makes FP abstractions such as functors and monoids a bit more accessible.
I can't disagree with the fact that learning category for that sole purpose sounds like a lot of work for too little benefit, though.
A category that captures the notion of sets and maps can be described in terms of arrows, objects and some semantic rules about which arrows (here interpreted as maps) connect which objects (here interpreted as sets).
Wiki on the "Category of sets [and maps]" -> http://en.wikipedia.org/wiki/Category_of_sets
Other rules about arrows between objects describe categories that capture other mathematical concepts.
Moreover for every object A, there is an identity arrow. Composition with that does yield the original arrow.
The point of those axioms is that they are satisfied by lots of different mathematical structures and idealized physical situations as well (Feynman diagrams, Chemical reaction networks, Knot-/Braid diagrams etc.).
For how this relates to computation, you can look at http://math.ucr.edu/home/baez/qg-fall2006/index.html#computa... for example. It is a bit more subtle than one would think at first, as in functional programming languages with "first class" functions, those are not not actually arrows, but objects. More precisely the type (A -> B) in Haskell, or (A=>B) in Scala, is a notation for an internal hom object.
The more interesting ideas arise when you then study how different such categories relate to each other, you then come up with Functors and how Functors between Categories are related to each other, those are called Natural Transformations. The original motivation for the invention of category theory was the study of natural transformations in homological algebra and algebraic topology. In the preceding years people had come up with several alternative functors (Cohomology / Homology on certain topological spaces) and they needed a way to talk about in what way those constructions were equivalent to each other.