Why there is weakness on the opposite side of the injury.

Movement of the body is primarily controlled by the pyramidal (or corticospinal) tract, a pathway of neurons that begins in the motor areas of the brain, projects down through the internal capsule, continues through the brainstem, decussates (or cross midline) at the upper medulla, then travels down the spinal cord into the motor neurons that control each muscle. In addition to this main pathway, there are smaller contributing pathways (including the anterior corticospinal tract), some portions of which do not cross the midline.
Because of this anatomy, injuries to the pyramidal tract above the medulla generally cause contralateral hemiparesis (weakness on the opposite side as the injury). Injuries at the lower medulla, spinal cord, and peripheral nerves result in ipsilateral hemiparesis.