There are a few things to consider before your podcast appears in the Apple directory. Therefore, please check the following points:  

• Is your podcast published?
• Does your podcast have a cover image? 
• Is the description filled out? 
• Have you already published an episode of your podcast?

Once you've done all that, the magic words "Publish to Apple Podcasts" should appear under "Edit Podcast --> Export".

Now the process can start:

• Click "Publish to Apple Podcasts". The link will take you to Podcast Connect, where you can submit your podcast.
• Log in with your Apple ID to manage your podcast
• If you don't have an Apple ID yet, you have to create one now at the latest, otherwise you won't be able to continue. You can get an Apple ID here: https://appleid.apple.com/account
• iTunes Connect will now check if your podcast meets all criteria. If yes, you will be taken to an Apple Podcasts page with information about your podcast. Check everything there, then click on "Submit" and wait until you are listed.

Some known issues:

Apple throws an error message: 

Please check again if you have really fulfilled all requirements (see above). Double is better!

The podcast does not appear: 

Apple sometimes takes a little longer to list the podcast. If you have created a new account, it can take up to two days until you can submit your podcast. 

Podcast Connect always throws me back to the login page: 

This is a known issue and unfortunately it happens more often with Apple. Try this workaround: subscribe to any podcast using Apple Podcasts, whether Mac, Windows or mobile. You just need to be logged in with the same Apple ID you used to submit your podcast. Then your Podcast Connect login should work.