Dogs with megaesophagus have an enlarged esophagus. This makes swallowing food difficult. Eating while in an upright position helps and the task of keeping the dog vertical is easier when the dog is sitting in a Bailey chair.
This blog post is less of a detailed how to and more of showing in progress photos. Adam, over at Lazy Guy DIY, has put together a great tutorial on how to build the Bailey Dog Chair. I followed his instructions with very few modifications.