By default RadioPublic will use mobile data when you are connected to a cellular phone or WiFi network. The amount of data used for each episode varies depending on the show. Some episodes use just a few MB whereas others might be 100MB or more. Episodes appearing with a "cloud" icon will use network data when played.
To reduce the amount of mobile data used consider downloading episodes while on WiFi. A downloaded episode will display a green and white down arrow. You can adjust how downloads work by visiting the Settings area within the app. It is common for people to have automatic downloads setup while on WiFi so they may have access to episodes while on the go without using their data plan.
The RadioPublic app will periodically look for new episodes or updates from shows when the app is open. Checking for updates uses a small amount of mobile data, and when new information about a show is available typically only a few KB of data is used for each show. You can allow the app to retrieve information about changes to shows when the app isn't open, this can be done in the Settings area for each app. If you want the most up-to-date episode information you'll likely want to enable background syncing.