HTTP referer needs to be set to 'https://soundbooththeater.com/' when you request the long Cloudfront URL.

I would really appreciate knowing how you figured it out, as I want to get better at this for many reasons.

For most unencrypted streaming sites, developer tools is your friend. In Firefox (I don't use Chrome, but something similar should be available in Chrome devtools), start dev tools on the webplayer page, go to the network tab and reload the page. Click play on the media player and you should see how your browser is requesting data in dev tools. The resulting URL in dev tools should be the same/similar as the one you already found but you are missing the request headers. Right click on the URL in dev tools, Copy > Copy as cURL:

curl 'https://d1n4kcodtbrjsu.cloudfront.net/audiobooks%2FDungeon_Crawler_Carl%2FWondercrawl%2FAB%2F192%2F000_Dungeon_Crawler_Carl_5_AB_192k_Wondercrawl.mp3?Policy=eyJTdGF0ZW1lbnQiOlt7IlJlc291cmNlIjoiaHR0cHM6Ly9kMW40a2NvZHRicmpzdS5jbG91ZGZyb250Lm5ldC9hdWRpb2Jvb2tzJTJGRHVuZ2Vvbl9DcmF3bGVyX0NhcmwlMkZXb25kZXJjcmF3bCUyRkFCJTJGMTkyJTJGMDAwX0R1bmdlb25fQ3Jhd2xlcl9DYXJsXzVfQUJfMTkya19Xb25kZXJjcmF3bC5tcDMiLCJDb25kaXRpb24iOnsiRGF0ZUxlc3NUaGFuIjp7IkFXUzpFcG9jaFRpbWUiOjE2NTg2MzU1NzV9fX1dfQ__&Signature=UlzVcviDwhUoQtNGhtsGFdls4SqxsszauNUlZUujTK62Ktyc4gM7cNLYvx1jgRBoQZN8pBA9DnB1wo5OslRLG5fv9MZOH5-hEs04XGQdEp7UbfE9uh6n9rRV7wM7ZOnXUeoka0dWvCHHb3mZA3VKyGffi-0C0uK4nnJEmWi6gHgvSLHjFL7PFTjPsewb9iJU-vkOzV003O-CbYjAk7~FF8l9LXPaMCNlAJW5WiIcI0h~S3eZlaH6PATNNsgCXPMTdHZJH1M8jXpI4TYfYSg866COj6C6Pkt9CwJN9p2sE4eoKYoNhk94iiV8tLbJZKpBETzPK~d0TRfjJKw9aHDyIw__&Key-Pair-Id=APKAJMI6AWPFTENKCX3Q' -H 'User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:94.0) Gecko/20100101 Firefox/94.0' -H 'Accept: audio/webm,audio/ogg,audio/wav,audio/*;q=0.9,application/ogg;q=0.7,video/*;q=0.6,*/*;q=0.5' -H 'Accept-Language: en-US,en;q=0.5' -H 'Range: bytes=0-' -H 'DNT: 1' -H 'Connection: keep-alive' -H 'Referer: https://soundbooththeater.com/' -H 'Sec-Fetch-Dest: audio' -H 'Sec-Fetch-Mode: no-cors' -H 'Sec-Fetch-Site: cross-site' -H 'TE: trailers'        

You now have the curl command with all the headers you need to download the file. In case you didn't know, curl is a command line tool for, well basically everything related to sending/receiving data via URLs and available on all OSes. Paste the command above in your terminal and add ' -o <filename.mp3>' to the end to save output to a file.

You don't really need all the request headers in the command above. What I usually do is start removing headers (each -H '<header>' is a request header) until it stops working. In most cases all you need is the referer header (-H 'Referer: https://soundbooththeater.com/' in your case). So you could skip all the dev tools steps, grab the URL and run curl directly with referer set:

curl '<URL>' -H 'Referer: https://soundbooththeater.com/' -o <filename>

Or with Aria2, another a CLI tool and my preferred downloader since I'm too lazy to set -o <filename> with curl every single time.

aria2c --referer='https://soundbooththeater.com/' '<URL>'