To properly play podcasts which are downloaded from Apple's Podcasts app, follow the following steps.

① Show the screen of a podcast you want to download.

② Push the "SUBSCRIBE" button to subscribe the podcast.

③ Tap the "My Podcasts" tab on the bottom of the screen to display My Podcasts screen.

④ Select the podcast from the list of My Podcasts.

⑤ Download an episode.

Podcasts can not be played, if you download them from a place other than "My Podcasts".

If the steps above doesn't work, please try downloading podcasts on your desktop computer and transfer them to your iPhone.