Well, if you did that the strings would keep ringing out. So you have to mute the strings you're not playing, and it helps to just not fret the ones you're not playing.

As for the muting, what you want to do (this is what I've found I do at least), is to use both your right and left hands for muting. Your left for muting the strings you've yet to pick, and your right for the strings you've already picked.