I don't necessarily agree with the premise that the insights that categories bring us are surprising and new. The framework is just formal and efficient.
In the 40's and later, mathematicans in fields between algebra and geometry were faced with a zoo of magical equivalences between abstract spaces and it was time for a clean up. With categories, mathematicans axiomize a fairly minimalist idea, namely arrow composition, and see where this alone takes one.
Now, for those who are interested, and to shine some more light on the statement you use to motivate your question (and to argue why "addition becomes multiplication" is not vodoo magic of the 21st century but combinatorically simple):
The diagram is associated with the coproduct, the abstract categorical motivation being the following: You start with two objects X1 and X2. You consider any third object Y at which X1 and X2 are both pointing at. The coproduct "X1 cop X2" in the middle is to be constructed by using two inclusion arrows i1 and i2, which lets you replace f1 and f2 by a single arrow f.
An application to mathematics: X1 and X2 are finite sets, like X1={a,b,c} and X2={y,z}, which contain 3 and 2 things, respectively. The arrows f1 and f2 are taken to be functions and they aim at a set Y as target. The introduced general object "X1 cop X2" must hence store the information what f1 and f2 might require, respectively: This is a set {i1(a), i1(b), i1(c), i2(y), i2(z)} containing arguments. The target to which these get sent to via f is Y. The list has 5 things.
"The cardinality of the coproduct is the addition of the cardinalities of the intial objects X1 and X2."
Here we have the product, the abstract categorical motivation being: You start with two objects X1 and X2. You consider any third object Y which points at X1 and X2 (notice arrow reversal). The product "X1 p X2" in the middle is to be constructed by using one arrow f (necessarily by the diagram, you have only one arrow!!), which lets you replace f1 and f2 by two arrows pi1 and pi2, the projections.
An application to mathematics: Again, X1 and X2 are the finte sets from above. The arrows f1 and f2 are now functions away from Y and one and the same thing from Y might get processed differently, depending on what f1 and f2 do. Here the introduced general object "X1 p X2" must be capable of storing the information from f, which express what arrows f1 and f2 can do for a Y arguments: This is a set which I'll write as {[fa,fy], [fb,fy], [fc,fy], [fa,fz], [fb,fz], [fc,fz]}. The list must store all the possibilities how targets of the arrows f1 and f2 could be combined. The list has 6 things.
"The cardinality of the coproduct is the multiplication of the cardinalities of the intial objects X1 and X2."
Notice that the difference merely comes from how many arrow heads end at the object which is to be defined: Can you use several lists which have index 1,2,... or do you have to make a grid? It's nice that plus turns to times by arrow dualizing, but the why is combinatorics which could have been explained to Pythagoras.
Now the formal defintion of the product and the coproduct, "the objects "X1 blubb X2" which is unique up to invertible arrow and which has the property that for any object Y, there are arrows..." looks terrible for any innocent person coming from outside. For the people working in a math jungle, it was a great tool for abstracting combinatorical principles away from the complicated spaces they were working with (not just finite sets like here).
You provide objects, you demand objects, and the relations (arrow, or and also functors, natural transformations, enriched cateogires, 2-categories, n-categories, adjunctions,...) represent the possibilies which then emerge. To make the point clear again: addition becoming multiplication by "arrow reversal" isn't a mystery. Here it's the difference between provinging one or two slots for saving information.
Don't be afraid of cats. The just capture everything and don't let you know.
What then happend is history. You can, similar to the construction for addition and multipliation above, capture whatever concept you like by "universal properties" (here an example of a diagram related to the product). People realized that categories are (somewhat clearly) the perfect tool for capturing order and operations on the whole construction. Subset relations (topos theory, see also this diagram), syntax and processes (logic, computation). Switching and matching them onto each other (functors, natural transformations).
As a sidenote: I find it interesting and also funny to keep track of names of mathematical subjects, which merge into each other or get rewritten. In relation to this answer, look how Wikipedians speak of this and that in the past, and this or this sound new and fresh - while the names of the subjects are not creatively very different.
Then, and this is the philosophical aspect of category theory*, you can replace equality be arrows too. "a=b" becomes "a isomorphic to b, and so b isomorphic to a". This way, you don't equate things, which are though of as different objects. The classical example being (I read it in a paper by J. Baez) that comparing two flocks of sheep can be done by comparing the sheeps one for one - or the two flocks can be compared to the sets of words "one, two, three, ...", which are invented for this purpose. Counting! The pro category argument is that the sheep get lost in the abstraction, and we should rather put the sheep in a category and not equate some cardinalities of flocks but construct the arrow putting the two flocks in bijection.
*apart from categories being able to capture logical syntax - which isn't surprising, as they just seem to be able to capture anything they want
I personally am not a fan of the numbers. The numbers as sets that is - I think the process of counting is the important thing. To answer your question though, I think here we're discussing just Platonism in sheep's clothing.
