It may ask for your password immediately after you tap the link. The Podcasts app will open and offer to subscribe. (In iOS 9, it will first open in News, before transferring you to the Podcast app). 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.
↑ / ↓ - increase / decrease volume
[ / ] - increase / decrease playback speed by 5%
j / l - 15 second jump back / forward
J / L - 60 second jump back / forward
space - pause
0-9 - Jump to that tenth of the podcast
Link to a time
Add a #time to the end of the URL, e.g. #356 or #5:56 to start 5 minutes 56 seconds into the recording