To go on top of what Ryan posted,
The cast is sent a box of playbills (lets say 25-50) and it goes around dressing room to dressing room and each cast member signs individually. There is usually a pen roped to the box if that trivia means anything to anyone

. The box is then picked up by the stage manager after a week or however long it takes, then the stage manager takes the letters, reads them and then sometimes posts the letters backstage or does whatever with them (haha probably trash after a while). Wicked released it's big fan book and published a few of them.
Jackie Hoffman who plays grandma in the Addams Family has a video blog, and she visits co-star Carolee Carmello (one of my favourite performers ever and one of my greatest fanmail successes) who is signing the endless box of playbills on film, if you actually want to see it. Not that exciting.
Jackie also reads fanmail outloud on her vlog, and on a few of them have asked the cast what their craziest or most bizarre fan letter said or who it was from. You can watch them on Broadway.com.
It gives you a great glimpse of backstage, and it's always funny when Nathan Lane pops up in costume or Bebe in her costume walking up & down the steps. It's also easy to see how annoyed people like Hoffman get when people send letters to her to have Nathan Lane sign. It's just rude.