With Mac OS X Catalina (10.15) onward, Apple introduced new security features, including Screen Recording permissions. This means that you will need to grant access to applications in order for screen recording to take place.
Whether you are using specific software, or a browser-based option (such as Podcast Studio) to record, you would need to ensure the relevant application is granted permission.
Granting Screen Recording permissions
Find the Screen Recording permissions by navigating to System Preferences > Security & Privacy > Privacy (as shown in the image below). Choose Screen Recording from the list. This will show you a list of applications which have requested screen recording permissions.
If you have not yet tried recording the screen, the application may not appear in this list, so you would need to open it first and attempt screen recording.
Once you have granted permission to the relevant application, you will need to restart it.