Answer:
First, let's recall some ideas that you may have:
1) The head of the girl pitches backward because of the inertia force, her body wants to be still, but the car moves, so the movement of her body is in the opposite direction of the movement of the car until her body is accelerated at the same rate that the car is.
2) The helium is less dense than the air, this is why the balloon floats.
Now, remember that the car is also filled with still air, this means that when the car moves, the air will also move backward as the head of the girl, and because the air is denser than the helium inside the balloon, the air will "displace" the air balloon in the direction opposite to its motion; if the air is moving backward, the air will impulse the ballon forward.
This is why the balloon pitches forward when the car accelerates.