I think there is something wrong with the problem. By convention the direction of the electric field is the direction of the force that a positive charge would experience if placed in the field. In this case no external force is needed to move the charge from A to B as it would naturally be accelerated by the field. Only the field does work.
If the charge had to be moved from B to A, an external force would be needed to move the charge against the repulsive force of the field. That requires positive work by an external force. At the same time the field does negative work because its force is in the opposite direction of the movement of the charge, taking the energy supplied by the external force and storing it as electrostatic potential energy. Then Yuvraj answer would apply.
The above said, there is another explanation for what the problem is getting at. If the charge is simply placed at A it will accelerate to B losing potential energy and gaining kinetic energy, as said above. But if at the same time an external force is applied to the charge equal to and opposite the force of the field bringing the charge to rest at point B, the charge will have moved at constant velocity from A to B and the overall change in kinetic energy is zero. Now the field does positive work and the external force does an equal amount of negative work taking the potential energy away from the charge.
The two scenarios have a gravity analogies.
In the first scenario, the charge moving from A to B under the influence of the field without opposition is analogous to the work done by gravity on a free falling object. The gravitational field does net positive work giving the object kinetic energy at the expense of gravitational potential energy.
The second scenario is analogous to taking an object at rest a height $h$ above the ground and slowly lowering it by applying an upward force equal to $mg$, bringing the object to rest on the ground, instead of free fall. The gravitational field does positive work on the object and my external force does an equal amount of negative work taking the gravitational potential energy away from the object.
Hope this helps