In the Podcasts app next to an episode. To download the episode so you can play it when you’re offline, tap > Podcasts, then do any of the following:
Prevent downloads from using cellular data (Wi-Fi + Cellular models): Turn on Block Downloads Over Cellular.
Keep your subscriptions up to date on all your devices where you’re signed in with the same Apple ID: Turn on Sync Podcasts.
Change the download settings for all your subscriptions: Change the settings below Episode Downloads.