In that part that says that the pulse is going from right to left with speed v, how does the air move from left to right at the same speed v ? is suppoused to be the opposite ?
The text says the pulse moves from right to left and you move with it at the same speed.
That means the air, which stands still in the rest frame, moves, in your reference frame, from left to right. Or, in other words, the not-moving air seems to move from left to right for you, because you are moving from right to left.