We enjoy it for the same reason that people enjoy playing Chess. It's a battle of wits and intelligence, to see who is the best strategist of a group of people, just like seeing who's the best player in an FPS, the best geared in an MMO, and so on.
Not to mention the fact that you can devise more cunning ways to defeat your opponent than just pointing a crosshair at someone's head and pressing Mouse-1.
Souplex said:
Every type of game evokes a feeling that of something:
Action: Skill/strength.
Stealth: Cowardice.
Turn based strategy: Intelligence.
Real Time Strategy: Memorizing build orders and keybinds.
Also, wow, you must really hate Stealth players. Or you must be absolutely terrible at being stealthy. Stealth isn't cowardice...if you've ever read a fantasy-esque book you'd know that discretion is the better part of valor.
Not to mention being a truly stealthy individual requires an absolutely brilliant and quick-thinking mind, years upon years of training and learning, and a knack for learning patterns, judging traits, and having a great deal of luck.
Action games requires memorization of combinations and how best to beat your opponent while not letting them counter-attack (and blaming your controller if you lose). Turn based strategy is all about planning. Planning, planning, planning, like in Fire Emblem. RTS...well, it varies. I can't call something like StarCraft an RTS with a straight face, so I'll go to something that's a real RTS: Total War. Total war requires you to make use of your mind in order to outsmart, out-think, and out maneuver your opponent. The third of these doubly true for games like Total War: Empires, and Total War: Napoleon. If you haven't got the mind for it, you'll be swamped under superior strategy and efficiency.