The reason you experience the chills when you spike a fever is the brain sets the desired temperature higher and as a result you perceive cold at your current body temperature. And also why you feel like you are in a sauna when your fever breaks, because the inverse is true.

To be clear this set point does mean induction of peripheral actions to actually elevate the temperature, but the central component of this symphony is the hypothalamus.

There is a lot of uncertainty around fever signaling, here is a great summary of what we know.