Here is my take on things.
Garen has been raised to respect the laws and customs of Demacia. The man will always fight for the greater good, no matter what that may entail. In addition, as a warrior, he is a person who looks for a challenge. Few people can truly match Garen's swordplay as far as combat. He's a real hero for Demacia, a leader of his own military force, and as such, he's reached a pinnacle of ability that few can match.
However, when Garen meets Katarina on the fields, he finds someone who is in virtually every way his equal. Their battles are the stuff of legend and tall tales on both sides. Each time she's spotted on the fields, Garen rushes to fight her, eager to prove himself as the better warrior between them, to test his strength against hers. He has incredible respect for her skill and mastery of her fighting ability. Even if Garen may not particularly like Noxus, he respects her as a fighter and combatant. Whether or not he finds her physically attractive is up to discussion, but any straight man can admit that Katarina is lovely to look at.
Katarina has been raised to respect strength and her own honor above all else, as well as her personal pride. When she fails, she works to rectify her mistake and prove herself as competent. She's very much a sort of poster child for Noxus despite her privileged upbringing. She's also proof that strength, in Noxus, if basically everything. She's one of the three best "blademasters" in Noxus: herself, her father, and Talon.
She meets Garen on the fields, and finds a foe that doesn't simply fall to her blows like the rank and file troops. He can fight back, hold his own, and meet and match her for skill and ability in his chosen craft. Because Noxus values and respects strength and personal honor and pride, Katarina respects Garen for his ability on the battlefield. She respects him for his strength and sense of duty. After all, Noxus does the same. Opposite sides or not, they form a sort of rivalry on the fields of war, fighting again and again to try and test their skills against one another. Whether or not Katarina finds Garen attractive is speculative, as I don't recall any comments from her about him.
In the end, hints of their supposed romance seem fanciful and teased, and I don't see any real reason WHY they should or WOULD fall into love, or become any kind of official couple. After all, let's face facts. Both respect their position in their city states, and to get involved with such a thing would most certainly be risky for both their positions. It just doesn't really work. Both are too married to their duties and positions to go about doing something like that.
Just my two cents.