Well, following along with the almost unreadable diagram....
http://chevy.oldcarmanualproject.com/electrical/wiring/images/54car1.jpgA "normal" 6 volt system works as follows: There are 3 pins on the flasher. One is power (apparently switched ignition power coming from a fuse mounted on the gas gauge on this car). One sends voltage from the flasher to the turn signal switch. The third pin is for an single indicator light, such as seen on the "Guide 6004" switch used on the 51 chevy and a few other things. Since there is only one indicator, and it blinks for either side, it has to have it's own separate flashing voltage supply from the flasher.
If there were 2 indicators instead of one, you would just connect each dash indicator to it's matching front signal. Not the rear signals, because if you connected the dash indicators to the rear signals, they would come on with the brake lights. You don't want that.
If you look at the diagram, the dash indicators are connected to the front signals as per normal, but, instead of grounding the dash indicator bulbs at the sockets, they ran them back to the third pin on the flasher, via a black wire. It really doesn't make much sense. Maybe it is supposed to do something odd to alert you when a bulb burns out?
It seems to me that in a left turn, both connections to the "left" dash indicator would be hot, and it would not come on. Meanwhile the right dash indicator would "find" a ground through the right signal light bulbs, and it would flash. I suspect Chevrolet just reversed the positions of the right and left indicators in the dash to make that appear normal.
It also seems to me that if you are not running a stock flasher due to a 12v conversion or whatever, you would have to ground the black wire (after carefully verifying it goes to the 2 dash indicator bulbs!) instead of connecting it to anything on the flasher. You would probably then have to reverse the positions of the indicator bulbs in the dash.