E.B.
My dd has been to two: one at around age 2 and one in her teens. Both were for food aversions due to medical problems, not anorexia or emotional problems, so I can't speak for those.
However, in the first one, the therapist (he was a psychologist) asked us a lot of questions, and just let dd play in the room while we talked. There were a lot of toys. There was no pressure on her. We explained how challenging it was to get her to eat (and we had a 4 year old at the time and he was a normal eater). The pediatrician had referred us to the feeding program because dd was so underweight.
After a couple of sessions he had dd sit at the table and observed what happened when food was set before her. That didn't go well. So over the next few sessions he would have her play while we ate some food at the table (some cheese and crackers, etc). She didn't have to approach the table. Then he had her sit at the table, but there wasn't any food. Then he had us bring in her brother and he sat and ate and we all talked.
Eventually he invited her to the table and set just a doll's plate with one Cheerio on it. She didn't have to eat it but she couldn't pitch a fit.
Gradually, she took a willing bite, although this took several weeks, even a couple of months.
Throughout all of this he would explain to us that her medical issues had caused an aversion to food, and he told us that the things we did with our son, who had no medical issues, just wouldn't work with our dd. It was very educational. I do think it was just as, or more, valuable for us, her parents, to figure out how to approach her.
The second time was when she was a young teen, and that too was worthwhile. The therapist talked directly to dd then, since she was older, and asked her what she did and did not like. Each session, a theme was presented, like fruits, or new tastes, or familiar foods in different forms (like pasta with a different kind of sauce), and dd had to sample them all. The therapist explored what dd did and did not like but got her to try new foods. It was pretty amazing: dd told her she did not like the taste of organic milk. The therapist did a blind taste test with 5 different milks (regular, organic, soy, skim, and goat) and dd correctly, and instantly, identified all 5. The end result was that she was willing to try some new things but we all agreed, there are some things she just can't handle. Bananas are completely out, and organic milk. Weird, I know.
But each time, I learned a lot. I didn't feel like dd was being allowed to not eat certain things out of stubbornness, but I felt like I was learning about her differences. And we learned how to introduce foods and what not to do. For us as parents, it was really eye-opening.