The effective mass concept takes all the complex internal forces due to the periodic potential, and sweeps them under the rug allowing you to relate an external force (eg an electric field) to the acceleration of a charge.
Consider an electron in a semiconductor under the influence of a field in the negative x-direction. The force on the electron is F=-eE, so the external force is in the positive x-direction.
Far from the zone boundary, the effective mass is positive, so by Fext = m*a, the acceleration is also positive. The electron accelerates in the positive x-direction towards the zone boundary.
As the electron approaches the zone boundary, the effective mass becomes negative (this can be seen from the curvature of the dispersion relation). The external force is still positive, which means that the electron acceleration now becomes negative, ie the electron decelerates. The negative effective mass tells you that the electron responds to the field opposite to how a free electron would.
Physically, the fact that the electron accelerates opposite to the direction of the force is because the electron must reflect off the zone boundary. As it approaches the boundary, it must decelerate. This behavior is of course due to the complex interaction with the periodic potential, but the effective mass serves as a convenient tool to understand how the electron will behave without knowing the details of these internal forces.P.S. In case you didn't learn it this way, a standing wave can be written as the sum of an equal forward and backward traveling wave, so it's just the same as saying the electron reflects from the boundary. And this may have just been a typo on your part, but it's the second derivitive and not the gradient of the dispersion relation that determines m*.
