I guess I caught a stange slice of a popular radio show. The official site is http://www.whitsend.org/ , not “www.atwitsend.org” as I thought from the broadcast. This makes much more sense. It’s a show called Adventures in Odyssey.
In fact, you can listen to the broadcast if you want–go to the daily radio archive pages here. The show I heard was the Friday 3/28 show, “Grand Opening, Part 2.” The part of the clip I listened to started at about 20:03 minutes in.
A quote from the clip: “How do I know you’re not part of the weird secret things?!”
Maybe it is a little bizarre at that.