I have installed a similar push button start in my 84 camper special. I know there is a neutral safety switch connection that is located under the dash that must be connected in order for it to start. This is what I used for my "on/off" switch. I'm not sure if its the same color wires as yours but it might be. Mine are yellow and purple (LSU colors). They need to be wired together in order to bypass this system and allow it to start.
Just to be sure here....but you are turning your key to the "on" position before you hit your start button....right?