Why?
Never in my entire life have I ridden a flawless bicycle. Inevitably, small imperfections in the drive train creep in. A flawless bicycle isn't flawless for more than a few hundred miles. All it takes is one whack on a bike rack, one hurried toss, one oops where the bike falls over as I'm grabbing my gels from the pantry. And never mind actual crashes! Ever broken a derailleur hanger?
The fixie keeps going basically no matter what. A problem for the fixie is the kind of problem that sends me to the hospital. Which is to say, that the fixie can take a lot more abuse than I can. Does the drivetrain wear out? Sure! And it costs basically nothing to replace!
Personally I think any serious cyclist should have one. They are super fun to ride, robust, cheap to work on, and they make you stronger. What's not to like?