Thanks for your interest in Podcast Chapters. If your question isn't answered in this FAQ, feel free to reach out to me on Twitter and I'll be sure to help you out.
The JSON export is new in Podcast Chapters 2.2. You can find it in the Podcast menu. When you export, information about your podcast episode is written in a standard JSON file into the folder you choose. Additionally, any artwork in your episode is also written to separate image files and refered to in the JSON data.
Export to JSON is very much the kind of feature you don't need if you don't know you do. The main point is to provide a way to use the information in your podcast episode in more places, places I as the app creator may never even imagine.
Perhaps you want to generate show notes in a particular format, use the podcast information to create a Tweet, put it in a database, or something completely different. JSON provides a reasonable format which is easy to work with for both humans and machines.
(By the way, I would love to hear what you use this feature for!)
You can upload the created files (podcast.js, any artwork images, player.html, player.css, and player.js) straight to a web server and use to let people listen to your podcast. You can also, of course, change the files any way you like to suit your needs.
For example, if you host your finished podcast files on a service such as Libsyn, you can change either the src property of the player, or the file name in podcast.js to point to Libsyn instead of a file in the same directory. If you link the player to Libsyn or some other host, your download and listening stats will be gathered as usual.
Another customization is of course to change the style to reflect your own site. Just alter - or throw away - player.css in any way you like. Everything is hopefully pretty basic. If not, please tell me what I can improve!