Not too sure about this, but I think this has got to do with the back EMF, generated when a current is induced in the wire.
When the motor first turns on, the current will be that of the one supplied by the voltage of the external circuit...
But as it starts spinning it experiences a change in magnetic flux, as it is moving within the magnetic field. Due to this change in magnetic flux, an induced EMF is generated, opposing the force that created it.
Let's say the wire was spinning anticlockwise, the opposing force would now be clockwise. Using the righthand palm rule, the force is going up [clockwise direction] and the magnetic field lines going to the right [NORTH magnet on the left and SOUTH magnet on the right], you can see that a current is generated in the opposite direction to that supplied by the external circuit.
It follows that, if the speed of the wire increases, the change in magnetic flux is greater. Therefore the strength of the backwards EMF must be greater, limiting the ultimate current the wire experiences.
In summary, at speed 0, when the voltage is run through the wire, the current supplied is that supplied by the external circuit. As the wire spins say at speed 1 on the x-axis, a backwards EMF is generating causing the current in the wire to be less than that of the original supplied one. As it gains more and more speed, the backwards EMF becomes stronger and therefore, the current supplied by the external circuit is limited, due to the backwards EMF generated. This causes the shape of the curve to look like that of A.
Maybe you can use this diagram of a motor to help you visualise it ^.
Hope this helped.