The “quiet” checkbox can be used to suppress notifications about actions taken by the owners on the list. 
By default, the list will notify the subscribers when they are added or removed from the list or when their settings are updated. In mass, this can result in a tremendous amount of not-very-helpful email being sent out automatically.
By checking the “quiet” checkbox, you turn off those notifications and the changes to your list happen behind the scenes.
This is the preferred option to reduce email traffic and confusion by your list members.
If you don’t use the quiet option, your subscribers will get a message like this when removed:
Your address (test@uakron.edu) has been removed from list list-name@lists.uakron.edu, likely due to excessive non-delivery reports for your address.
Which is neither helpful nor accurate.