I spent a while looking for this solution - I found it on a Reddit thread so I'm sharing it here for anyone reinstalling macOS 10.11 El Capitan:
1. In Safari, go to https://audiocontentdownload.apple.com
2. In the "Safari can't verify the identity" dialog, click "Show Certificate"
3. Enable the "Always trust" checkbox
4. Click Continue
5. Relaunch GarageBand and retry downloading
The audiocontentdownload.apple.com certificate is too new for OS X 10.11, so this adds an exception to your Keychain that allows it to be trusted by Safari and other applications.
I tested this and now the sounds (essential and the rest) are loading properly in GarageBand. Previously, it would stall out and fail to load anything.
1. In Safari, go to https://audiocontentdownload.apple.com
2. In the "Safari can't verify the identity" dialog, click "Show Certificate"
3. Enable the "Always trust" checkbox
4. Click Continue
5. Relaunch GarageBand and retry downloading
The audiocontentdownload.apple.com certificate is too new for OS X 10.11, so this adds an exception to your Keychain that allows it to be trusted by Safari and other applications.
I tested this and now the sounds (essential and the rest) are loading properly in GarageBand. Previously, it would stall out and fail to load anything.