Google Podcasts started with a bare-bones interface and even more bare-bones functionality. However, over the past months, it has slowly but steadily been adding some trequisites, such as playback speed, sleep timer, silence trimming, Auto support, a web interface, episode search, as well as one particularly special feature based on full transcriptions of shows with keyword search suggestions. But today, we're talking about an addition that fits in the former category: auto-downloads.
Yes, after over seven months in the making, Google Podcasts is finally adding an option to auto-download new episodes from podcasts. You can disable this completely, enable it only on WiFi or also on your data network, and choose which podcasts to auto-download and which not to. There's no scheduling though, or a way to limit this to only when your phone is charging, so be wary of how many podcasts you enable it for as it could kick in at unexpected times during the day.
The feature has been showing up for some users for a month, though it seems to be a server-side switch. It's possible you've had it for a while, or you just got it (like me), or maybe you're still waiting for it to reach your device. To check it out, go to Settings in Google Podcasts and look for the Auto Downloading item at the top.
You can grab Google Podcasts from the Play Store, but keep in mind this is mostly just a shortcut to the feature, which resides inside the main Google app.