Plex announced podcast support a few months ago, but some admittedly basic features were missing. In Plex's defense, podcasts were still in beta. Today, podcasts leave beta and get some of those basic (and vital) features.
Here's the full changelog for the new update.
[Mobile] Android Auto support for Podcasts.
[Mobile] Podcasts: improve artwork.
[Mobile] Now you can download podcasts, including immediate download of episodes or set up automatic download for shows.
[Mobile] Downloaded podcasts can be viewed and played when no internet connection is available or when switched to offline mode.
[Mobile] Import your current podcasts subscriptions from other apps via the Import OPML action.
The biggest change is offline support—you can actually download podcasts to your device rather than stream them. You can choose how many episodes you want to keep on your phone and whether or not they should be deleted after you listen to them. You can also play those podcasts with Sonos speakers, cast devices, and in your car with Android Auto.
If you're switching to Plex for your podcast listening, you don't have to search for everything and add URLs manually anymore. With the latest update, Plex supports OPML imports. Just point it at the file, and your podcasts will be ready to go.