Why pick master yi? Because hes fast, hes that guy that makes you feel like youre standing still even though youre running, because he runs and attacks so fast, its like hes just there attacking you even though youre actually running for your live. He has a powerful steroid that resets upon killing, and makes him unslowable, he has a sort of shield that will protect him against any powerful nuke by increasing his resistances to godlike levels. Whats more, his gap closer can be used to avoid damage, and allows him to have one of the fastest jungle timers. His ganks are powerful, not many people can escape a master yi with ult and red buff, not without wasting atleast two summoner spells.
Im not saying Master Yi is better than say Nocturne or Xin Zhao, but he does have reasons to be played, I personally find Master Yi more enjoyable than those guys and would still not pick him or want him in my team for a ranked game. Hes really fun, it feels like going ALL OUT when you ult and activate ghostblade to kill someone, its really fun. Other AD junglers dont feel the same way as they dont have an ult to increase their movement speed so much.