It's not a podcast, it's a live radio stream, but indeed it has a fixed position at the bottom. That's easy to achieve with CSS. Watching the website more closely when you click on a link, you can see that not only the audio player stays in place, all other content is static too, except the main content column.
It's a bit technical to explain, but this can be done with AJAX. Meaning when clicking on a link the content is requested in an ajax call and then loaded into the main div without refreshing the page. If you're familiar with coding, this can be a starting point:
Load content in a div with ajax.