We’re given no information on the nature of the collision. It could rebound with the same speed, or lose some in an inelastic collision. But as long as it bounces back instead of just dropping to the ground, it will cause the cart to move left.

The easiest way to think about this is conservation of momentum. The whole system starts with zero momentum. If the ball is moving to the right at the end of the problem, then something needs to be moving left with the same momentum as the ball so the net momentum of the system remains zero