RadioPublic uses storage on your phone when episodes are played or downloaded. Episodes are automatically removed after a period of time once they are heard. You can adjust how this works in your Settings. You can also manually remove an episode by tapping the "Remove download" from the episode download menu. You can free up storage space by removing all downloaded episodes from the Settings.
You can also set things up so the RadioPublic app will automatically download new episodes for you. Typically people enable automatic downloads to ensure new episodes from shows they enjoy are available without having to use their mobile data while not on WiFI. You can adjust how downloads work in the Settings area for the app.
Manually downloading episodes on iOS