This particular riff, as well as other parts in the song, utilize a lot of palm muting, power chords, and octave chords.
All of the octave chords are played on the A and G strings. When playing these types of chords, make sure only the root note and its octave are sounding, so only the A string and the G string should sound. Use your fret-hand index finger to mute the remaining 4 strings.
Now since we're in Drop D tuning, our power chords will be different from how we'd normally play them with standard tuning. It's actually a lot easier to play Drop D power chords. Take a look at the 4th measure in the tab below. See how all the notes in each chord are in the exact same fret? So all you have to do to play Drop D power chords is place your index finger over the appropriate notes in a single fret.