Apple only allows downloads to play on your iPhone through iTunes and the Podcasts app. You can click on the live player and stream the live show through your mobile browser. But if you are seeing a black screen with an audio icon with a line through it, that means you have tried to download the show through your mobile browser, which Apple doesn’t allow.
To listen to a download of the show on your iPhone, you will need to add the show to the Podcasts app on your iPhone. Just follow the instructions to add the show to the Podcasts app if you haven’t done so already.