It’s like unfollowing someone without actually unfollowing them.
Mute simply means a user’s tweets and retweets will no longer show up in your timeline. Just go to the user’s profile, click on the gear button and mute @username. You may unmute a user at anytime by performing the same action.
Lucky for you, a user will not know that they have been silenced.
As with most new features, the mute button is being rolled out to users over the next several weeks.