I installed a speed control switch in my ceiling fan. Now when I pull on the light switch chain when the lights turn off so does the fan. Why is this

If I were venturing a guess I would say that you connected the fan switch to the light wiring. Instead of connecting the fan switch to the wire providing power to the fan, you connected it to the wire providing power to the light and you connected it after the switch to the light. Lots of wires in there and probably easy to get confused. You should have an unconnected wire in there some where and that's probably the fan power supply wire.

I am assuming this is a replacement switch.