Why does our left hemisphere of brain control our right side of our body and the right our left?

Because of a phenomenon called Neuronal Decussation.

Our bodies move and work through muscle contractions. Those contractions are facilitated by neuronal impulses delivered by motor neurons that arise from the spinal cord (To be more specific, the anterior horn of the gray matter in the spinal cord). The anterior motor neurons are also called lower motor neurons because they are downstream the neuronal circuitry, below the first synapse.

These anterior horn motor neurons in the spinal cord receive information from the higher centres (the cerebral cortex) through upper motor neurons. It is because of the crossing over of the upper motor neurons to the opposite site along their course that the right hemisphere controls the left side of the body and the left hemisphere controls the right. This crossing over is known as Decussation. There is little literature on the evolutionary benefits conferred by this organisational paradigm.

The upper motor neurons predominantly include the Corticospinal Tract and the corticonuclear fibres which together constitute the Pyramidal tract.

The Lower motor neurons are motor neurons that arise from the anterior horn cells and the twelve cranial nerves. The former supply most of the body while the latter supply the head and neck.

The decussation of Corticospinal Tract occurs at the lower part of the medulla.

Decussation is not unique to Upper motor neurons. Cranial nerves like Trochlear nerves also decussate.

The decussation of the trochlear cranial nerves occur just concomitant to their emergence from the midbrain. They are the only cranial nerve to do so.

Reference: Udayabhanu Bhanja