When I was a young girl, I sent a fan letter to Linda Davis, the country singer. You may remember her? She sang the duet "Does He Love You?" with Reba McEntire.
She wrote me back.
How do I know it was her?
Well, I went to a Reba concert a few months later, and Linda stayed out in the lobby after the concert to autograph things. I walked up with a shirt, and, as she was signing it, I told her about me letter. She remembered me.
I treasure both the shirt and reply letter to this day. It was really something to connect in some small way to someone that I really liked as an artist.
In fact, if I thought the likelihood was high Kate Mulgrew actually read her fan mail (which she says she does, in fact, read all of it), then I might send a letter now. But, I think it's far more likely her assistant opens it, reads it, scans for interesting stuff, and sends that over to Kate for later reading. So... meh...
Anyway, my point is that I think people do it so they can feel connected to a person they would otherwise never have a chance to connect with. You know?