Is it impossible to listen to podcasts on iOS 6 now?
If you had downloaded the podcasts app while iOS6 was the most recent, then you would have a version of the podcasts app that was compatible with it.  Unfortunately the current version of the podcasts app is written using iOS7 APIs, including ones to automatically download content for  you without you having to manually open the app etc, which are not backwards compatible.

As such, you will have to use a 3rd party app to listen to podcasts, or upgrade to iOS7.