This part has a lot of 16th note open notes on the Low E string. And mixed in between we'll have some 8th note rests and quarter note rests.
Notice how my left hand is muting the strings again during rests? Be sure to always do that.
When you play the octave chords with your index and ring fingers, make sure you mute the strings you aren't playing. In this case, every string except the A and G strings should be muted when you play the octaves. The best way to mute them is to have your index finger resting on them while playing.