belching is another way of saying burping: That thing you do after drinking or eating when you make noise come out your mouth. "belching lessons" just means that someone is teaching him how to do it.

Another word for belching is "burping." Both mean "to expel gas noisily from the stomach." Phoebe, Holden Caulfield's sister, is ten years old. She is behaving like children of that age, who are openly interested in bodily functions. Belching is usually consider to be impolite. Belching normally occurs as a byproduct of eating, particularly eating too much or eating too quickly, or drinking carbonated beverages. However, there is also a trick, that can be mastered, of intentionally swallowing air without eating or drinking anything, so that you can, if you wish, belch frequently, or make large belches. I had a friend in college who could do it, and enjoyed _talking_ in belches. That is, he would swallow a lot of air, make a long, loud belch, and form words while belching. Belching is considered impolite, but children often like to do impolite things. Phoebe's friend has mastered the art of swallowing air to produce large, loud belches. Phoebe thinks that's really cool and wants to do it, too. Her friend is giving her lessons in how to do it.