Need to do something else on your phone? The podcast will continue playing from the Google app if you leave, and those controls are easily accessible by swiping down on the notifications menu -- similar to other audio apps. What's more, some podcasts that aren't available through Google Play Music are with this new feature in the main Google app. For example, searching for Serial or Fresh Air produces the last few episodes, but those two popular podcasts don't show up in Music's web-based desktop app. The opposite is true as well: some of the podcasts in Music on the desktop don't immediately pop up when I search for them in the Google app.
The initial search results show you the three most recent episodes with a play button beside each one. If you need to go back further, tapping "more episodes" will display the podcast's description and a longer episode list. In the case of Serial, you see all 26 episodes, but for one that's been around longer, like Fresh Air, Google will let you browse the last 50. If you don't mind streaming podcasts, this is a handy add-on to an app you already have on your phone. However, if you prefer to take your listening offline, you'll need to look into options like Pocket Casts, Spotify and others.