Like attracts like, and in this case they're both the same 'type'. They both enjoy the old ultraviolence, it's just that Katarina was raised in a culture that not only accepts a killer instinct but condones it in the name of Noxian supremacy. Garen was raised in a society that frowns on that kind of behavior in general but praises it in their military heroes, so he naturally channels that into activities his culture finds praiseworthy, such as cutting his enemies in half (for Demacia of course).
If Lux's league judgment is anything to go off of, Demacian society sounds very regimented and apparently they send all their kids to military 'brainwashing' school at a young age to learn how to be properly Demacian. I can see how a person like Garen who is essentially the same 'type' but who lives in a very ordered society would be attracted to someone like Katarina, and it would go on multiple levels.
1. omg hot redhead
2. Wow, her blade technique is flawless and she parried my attack, I am actually possibly in danger of being killed for the first time in my life, this is making my heart race and I have never felt like this before
3. Her style of life and her culture is diametrically opposed to my own, I find that strangely compelling
Katarina isn't a backroom double-dealer (she frowns on the kind of tricks her sister uses, for example, and dislikes Swain) and being raised Noxian she probably has a healthy respect for skill and power, and Garen is a powerful and handsome man she has been unable to kill several times. Throw in the charm of having a 'rival' who seeks you out constantly on the battlefield for pulse-pounding non-stop lethal combat acrobatics and you have some fascinating date material.
You don't really need to present an argument for it because the only conflict between them is in their allegiances. They both are very good at killing, they both enjoy fighting, they're both members of famous families with vested interests in promoting their respective city-states.
Also, Blitzcrank rated their compatibility 100% and I'm just a simple fleshbag; who am I to question?