RadioPublic uses storage on your phone when episodes are played or downloaded. Storage is cleared up automatically after a period of time once episodes are heard. You can also manually remove an episode and free up storage by using "Remove download" from the episode download menu. You can free up storage space by removing all downloaded episodes from the Settings.
It's also possible to allow the RadioPublic app to automatically download episodes on your behalf. Typically people will enable automatic downloads to ensure new episodes from shows they enjoy are available without having to manually download them. You can adjust how this works in the Settings area for each app.