The displacement of a body is proportional to the cube of time elapsed. The magnitude of acceleration is increasing with time. Why?

have you leaned about derivatives yet? Specifically: the derivative of the displacement with respect to time gives velocity, and the derivative of velocity with respect to time gives acceleration. If you have not yet been exposed to calculus then I suggest drawing a graph of a x = At^3, and see how the slope varies with time (to get the velocity graph), then how the slope of the velocity graph varies with time (to get the acceleration graph)