macOS imposes privacy restrictions that disallow, by default, access to certain application data (e.g. Mail, Messages, Safari, Photos), and all access to any external hard drives and network volumes.
macOS does not conveniently ask you to grant access to an application when that application tries to access that data. Instead, you're required to manually pre-approve the application. As a result, when you download an application specifically to back up your most precious data, that application can't back up that data until you specifically go out of your way to grant it access to that data.
To proactively grant full disk access to CCC, choose "Grant Full Disk Access to CCC..." from the Carbon Copy Cloner menu.
Troubleshooting Full Disk Access problems
If you have granted CCC Full Disk Access, and your backup tasks still report that CCC lacks Full Disk Access, examine the configuration of any antivirus software that you have running on your Mac. The Device Control feature in ESET Cyber Security, for example, can prohibit access to external storage in the same manner that a lack of Full Disk Access would prohibit CCC from accessing external storage.