How do I link a podcast to a website URL to show up in Google's rich search results?

Include the URL of your podcast in your RSS feed, which can be done when you generate the feed. Also, on the target web page, place a special meta tag (alternate link) in the section that links back to the URL of your RSS feed. With this cross-linking, Google will be able to discover and link your URL to the podcast.

<head>
  …
  <link rel="alternate" type="application/rss+xml" title="Podcast Title*" href="Feed URL**">
  …
</head>

* "Feed Title" is the title of your podcast,
** "Feed URL" is the URL of your RSS feed.