Assign the first of the 6 consecutive integers as x.
Set I contains: x, x+1, x+2, x+3, x+4, x+5
Set J contains: from adding 3 > x+3, x+4, x+5, x+6, x+7, x+8
And also from subtracting 3 > x-3, x-2, x-1, x, x+1, x+2
Only x+6, x+7, x+8, x-3, x-2, x-1 from set J are not in set
I, so there are 6 more integers in set J than in set I, which is answer
choice D.