Hosting, analytics, and distribution tools for your podcast
Nope! Apple does not host your podcast for you. It's just a directory.
Every podcast needs to purchase their own hosting for their MP3 files, to generate a valid RSS feed, and show notes.
iTunes, Apple Podcasts, Spotify, Overcast, and Stitcher are podcast players that also feature a podcast directory. But they're not hosting platforms. When you click "download" or "subscribe" in these apps, the request gets sent to whichever host your favorite podcast is using (like Transistor.fm!). The podcast player will then download the MP3 file off that server.
If you've ever clicked "download" for a podcast, and it didn't work, it's likely that podcast isn't hosted with a reliable provider!