Why does your human skin feel intensely cold and develop shivers right as a high fever temperature is actively climbing?

Your internal biological thermostat is managed by the hypothalamus in the brain. When a virus invades immune cells release pyrogen signals that force the hypothalamus to artificially raise your target core temperature goal to fight the infection. Because your current body temperature is still at a normal 37 degrees your brain misinterprets the gap as being freezing cold.