Silencing the Chords is an important part of playing the Venezuelan Cuatro. It gives you the ability of playing with more elegance and gives a great percussive feeling to the music you are playing. A great way to start off your technique as an intermediate Cuatrista. As professor Toro suggests, practice this with chords making use of all 4 fingers and just jam along as you wish and get a better feeling for applying this technique (subtitles on for the video).
The way in which you can block the sound to give this effect is to lift your left hand (or the hand you are using to mark the chords) and gently touch the strings. The resulting effect is that of a percussive one as is demonstrated in the video.
Although this may seem easy, it needs to be carefully executed to get the correct effect. Marking this is necessary to maintain the rhythm and it also gives an extra percussive effect. This is constantly used by cuatristas to enrich the music they play.