Here are the most popular free and inexpensive sources for royalty-free music, DIY loops and sound effects.
Royalty-free audio is created expressly so you can use it legally, without having to pay ongoing royalties or license fees. You must use only “royalty-free” music in your podcast, or music that you otherwise have permission or an explicit license to use.
Note that “royalty-free” (i.e. free to use) doesn’t neccessarily mean “free” (i.e. free to license), although some music is both.
Please pay attention to the Terms and Conditions on anything you download, and make sure you provide proper attribution when needed. (Tip: Show notes are good for this.)
Adobe Audition Downloads — Over 10,000 high-quality sound effects and 1000s of high-quality loops. These are austensibly for Adobe Audition users, but are available for everyone to download and use.
Free Music Archive — A library of high-quality, legal audio downloads directed by WFMU, the most renowned freeform radio station in America.
Freesound — Collaborative database of audio snippets, samples, recordings, and bleeps, all released under Creative Commons licenses that allow their reuse.
Music for Makers — Give ’em your email address to get one royalty-free song in your inbox every Monday. (You can also give them $9/month for access to all of their tracks.)
YouTube Audio Library — The YouTube Audio Library is a free collection of music and sound effects for your podcasts. Tracks with "attribution-required" icons are Creative Commons-licensed, and require that you credit the original artist in your show or episode description.
GarageBand (macOS, iOS) — If you want to try your hand at some original music, GarageBand is a great way to do it. Use the GarageBand > Sound Library menu if you want all of the available loops (more than 1,000).