Why do I get my old MS symptoms back when I am heated up?

Nerve transmission. This is a well known phenomenon and was used in diagnosis in the past. It is assumed to be caused by heat slowing conduction in previously damaged nerves.
Old MS pathways. Electric signal transmission across nerve fibers (axons) requires good insulation (myelin). While MS inflammation is followed by myelin repair, the extent of repair may vary and is never perfect. When core body temperature goes up, some of these old MS lesioned pathways are prone to "drop the signal". This is the "uhthoff phenomenon". No permanent damage is done.
Insulation. Your myelin nerve coverings are partially dysfunctional due to prior inflammatory attack and heat of any kind can cause changes in intracellular potassium resulting in symptoms. This is reversable by cooling.