The chord is a D minor 7 (Dm7). This chord can be played a number of ways, but in the video he is barring the 5th fret and playing the Am7 form. In this form, you would not typically play the 6th string.
The 5th and 6th strings are NOT open. Your index finger is used as a "barre" - it frets all of the strings at the 5th fret. Then you use the other fingers to play a transposed Am7 chord. Normally this is played as follows:
6th string - not played
5th string - barred at 5th fret with index
4th string - 7th fret with ring finger
3rd string - barred at 5th fret with index
2nd string - 6th fret with middle finger
1st string - barred at 5th fret with index