Ordinarily, students are notified when you grade a Canvas assignment. The mute setting keeps students from seeing grades and comments on any assignment it is applied to. This feature is therefore useful if you want to delay releasing grades, or if you want to make sure grades are released for everyone at the same time. You can mute an assignment in the Gradebook or in Speedgrader.
You can use the Student View to see what a muted assignment looks like from a student perspective.