Personally, I think it was less about it making actual sense (like Water being good vs Fire for very obvious reasons) and more that Game Freak wanted to balance the game.
As of earlier, Figthing types are quite powerful, with their attacks being Super Effective against 5 other types (Rock, Steel, Ice, Dark, Normal) and only be resisted by 4 quite common, quite frail types (Poison, Psychic, Flying, Bug).
And they are not ligthweigths on the defensive front either. Psychic and Flying are relatively rare attacking types, and Figthings resistance to Bug, Dark and Rock is no laughing matter.
Then there are the Figthing types themselves. Lucario, Machamp, Conceldurr, Mienshao, Hitmontop, Scrafty, Medicham, Blaziken, Infernape, Hariyama, Heracross, Terrakion, Cobalion, Virizion, Toxicroak and Keldeo are all powerhouses, and some dominate the competitive battlefield.
Based on that, Fighting type was almost in need of a nerf as the Dragon type.
That, is my theory to why Fairy resist Figthing moves and Figthing types being weak to Fairy.
However, if you want some more logical explanation, it could be that Fairy types are so cute the big bad Fighting brutes don't want to pummel them.