How to add podcasts?
You can subscribe to podcasts in Feedly but the playback experience is not optimized (ie. playblack in background is not supported, downloading podcasts, etc.).
You can add podcasts by subscribing directly to their RSS feed and add them to your feedly via the plus button in the left sidebar.
Some 3rd party apps are better managed for podcasts and integrate with Feedly, so you can try them as well: