Go to Settings and ensure you have "Force offline" off, and you're set to update and download when on wi-fi (assuming you are regularly on a good wi-fi connection).
In Settings > Downloads, ensure your Subscriptions and Play Later have limits above zero.
Open an episode you wish to download (tap on its title to open the episode detail screen). In the 3-dot menu, tap on "Links" and then navigate to "Raw episode file" to see an option to open the raw file URL. Try this and see if it plays. If not, it may be an issue with the publisher's servers.
Try to log out and log back in. It shouldn't be necessary but sometimes fixes the problem. Feel free to go into Settings > Help > [Contact us] if you'd like us to investigate any problems with downloads.
Why aren't downloads working?