Google Play Music Podcasts is only available in the United States. You will not be able to submit a podcast if you are located outside of the US. Some users may have success with a US-based VPN however it is not officially supported.
Unlike Apple Podcasts, Google Play Music caches your podcast episodes on their own servers meaning that you don't get any analytics data about plays in Google Play Music. Dynamic ad insertion is also not possible.
To submit your podcast to Google Play Music:
Click “Get Started.”
Login with your Google account.
Click “Add a Podcast.”
Review and accept the terms of service.
Enter your RSS feed, you can find your RSS feed under Playlist "Details".
*IMPORTANT* You need to have an email address in your RSS feed to submit to Google Play. In Omny, this can be updated at program settings - details - contact information (screenshot below). Your feed will be rejected without this field entered.
Click “Publish Podcast.”
You will receive email approval once your feed is live!