It may ask for your password immediately after you tap the link. The Podcasts app will open and offer to subscribe. After you confirm, it may again ask for your password before the podcast will be added to My Podcasts.
Other apps on your phone or computer:
Search for "Great podcast apps for my iPhone/Android/Mac/PC" to find other ways to listen.
If that link does not work, your app may require you to copy/paste the address. Right click the RSS Feed link and select Copy Link. On a mobile device, tap and hold to bring up a menu that will allow you to copy the link.