The loss of sensation on the opposite side of the lesion is because the nerve fibers of the spinothalamic tract (which carry information about pain and temperature) crossover once they meet the spinal cord from the peripheries.
As a result of the crossing of nerve fibers in the spinothalamic tract, there is a loss of sensation on the opposite side of the lesion. This occurs when the sensory information from the periphery crosses over to the same side of the spinal cord before being transmitted.