Muting simply means to stop the string from vibrating, or sustaining it's note. It is very simple to do with a little practice.
What I mean by quarter note is the down beat of the song. When you are taping your foot to the beat of the music, that's counting a quarter note. The eight note is the up-beat or the notes in-between the quarter notes.
You can practice it a little slower by doing all quarter notes (or down beats) On the first down beat strike the E string and on the second down beat, bring your left hand down on the E string and stop the string from vibrating, do not fret a note, just stop the string from vibrating. Then do the same with the A string. Try to keep a constant rhyme and alternate between the A and E string. Once you get it down solid, move to quarter notes and muting the string on the eight note (up beat)
Playing open string positions are common and learning to mute the strings is something you really can't avoid. Not learning proper technique and cutting corners to compensate for the lack of technique will only hurt your playing in the long run. Why? it is far harder to fix bad technique by relearning to play properly than it is to learn to play correctly in the first place.
Practice, Practice, Practice...... If it was easy, everyone would be doing it.
