Sign in to the app with your Slate Plus account (more info here)
Select the Menu icon from the main screen (three parallel lines), then select Podcasts. Then tap Shows.
Tap the show you want to listen to, then you'll find a list of the latest episodes. If you are having trouble with an episode in the Shows section, you can also try under the "Slate Plus Podcasts" icon, which lists all Plus podcasts by release date.
*Please note that though the episode timestamp may display the shorter length at first, when loaded the podcast will play the full extended Plus version.
To learn more about accessing your Slate Plus podcast benefits outside of the Slate iOS app, in the podcast player of your choice, follow the instructions here.