How is it structurally explained that the motor activity of the left side of the body is controlled by the right cerebral hemisphere and the motor activity of the right side of the body is controlled by the left cerebral hemisphere?

In the cerebral hemispheres there are neurons that centrally command and control muscle movements. These neurons are known as superior motor neurons and they are located in a special gyrus of both frontal lobes called as motor gyrus (or precentral gyrus). The superior motor neurons send axons that transmit impulses to the inferior motor neurons of the spinal cord (for neck, trunk and limb movements) and to the motor nuclei of the cranial nerves (for face, eyes and mouth movements).

The fibers cross to the other side in exact areas of those axon paths. About 2/3 of the fibers that go down the spinal cord cross at the medullar level forming a structure known as pyramidal decussation. The other (1/3) of fibers descend in the same side of their original cerebral hemisphere and cross only within the spinal cord at the level where their associated motor spinal root exit. The fibers that command the inferior motor neurons of the cranial nerves cross to the other side just before the link with the nuclei of these nerves.