Here's a breakdown:
* Utter: To speak or say something.
* Nary: Not any; none at all.
* Sound: A noise.
So, "utter nary a sound" literally means to "speak not any sound", which is a more emphatic way of saying "be completely silent".
Example:
"The room was so quiet you could hear a pin drop, and everyone sat motionless, uttering nary a sound."