As long as you have "browser" selected as the device for answering the calls in LiveAgent it also needs to have access to your microphone for you to be able to make or accept calls. If the microphone is blocked or simply not accessible by LiveAgent you'll receive an error with "Call failed. Reason: User Denied Media. Originator: local".
Normally when you accept a call or make an outgoing call in LiveAgent, your browser asks you for permissions to use your microphone which must be granted in order for LiveAgent to work correctly.
If by any chance you deny using your microphone you might not be able to answer calls or create new ones in LiveAgent and would experience unexpected behavior.
Therefore it is a good idea to always allow usage of your microphone. This must be done in browser and also in Windows itself if you are using Windows 10.