I'm trying to remember how these cigarette lighters operate, but here is how I read the diagram. My thought was without the lighter inserted, there is no ground (essentially a dead 30 post) or basically a break in the shown lighter ground wire and the 30 post wire. Insert the lighter and the ground path from the shown lighter ground wire goes through the 30 post, through the 87 post, then onto the next relay 85 post, through 86 post, then back onto the lighter relays 86 post and through the 85 post to the relay's ground wire. Without that continuous path provided by the inserted lighter, the lighter relay contact doesn't energize and will be stuck on the 87a 12V being passed to the 30 post and onto the lighter.
Now how the lighter itself operates I can't recall....been forever since I've even looked that those things!
I believe the lighter would have to be pushed in as if you were getting it heated up for use.
Also, key side starter is what is attached to the 30. This is basically the wire at your ignition switch that goes to the starter solenoid (brown I think??), in which you would cut and put one end on the 30 and the other on 87.
Edit: I think the lighters work as a bit of a short circuit, as in when they are pushed in the 12v wire and the ground get connected with a copper resistance between, which heats things up. So by pushing in the lighter in this instance, the ground path gets completed. I'm not sure how the lighter ever operates though with this diagram.