I would say that's situation dependent. Sometimes you can let the note ring and don't need to silence it. When I need a more staccato note, sometimes I lift my fretting finger just enough that the string no longer sounds (finger still in contact with string, the string just comes up off the fret). There are also situations in which I'll explicitly move a left handed finger to mute the string, or mute it with my right palm or something like that. I go with whatever feels the most natural and relaxed, something that won't hamper my playing or alter my technique. I also try to use a method that fits the feel of the piece I'm trying to learn, as the different techniques will sound different.
In the end, I would caution learning only from a book, and this is one of the reasons. I've certainly picked up a technique or two by reading, and certainly lots of theory, but some books are written with the assumption that they don't need to explain some things. For me, learning to play music is one of those things that benefits greatly from another pair of eyes, another pair of ears, and/or the ability to follow the example of someone more experienced.
That being said, if you let us know what exactly you're playing, someone here might be able to offer a specific suggestion. The answer will differ if you're reading about power chords v. a pentatonic scale.
A quick YouTube search brought up a video in which the guitarist addresses muting the strings. He is muting with his strumming hand in this case. If you're playing something more like the flute melody described in this tab, I would simply mute by lifting my fretting finger slightly, since I'm playing the next note with a different finger.