The question is, how much current is flowing and is it too much. If you have a standard coil (a non-ballast, nominal 3 Ohm coil) then closing the points and/or shorting that wire to earth will only allow about 4 Amps to pass. That will make the wire warm but it should not melt. I doubt you have a ballast coil installed but if you did, and if that coil was running without a ballast resistor, it would allow too much current to flow through the distributor and then the wire could get quite hot.
For future reference, do not use just any bit of wire to replace the one that burned out. It is supposed to be a high-flex wire that allows free movement of the breaker plate under vacuum advance. Buy a new proper lead or visit a motor shop and buy some used brushes with similar wire that you can salvage and solder in place of the melted one. Coat that wire with a bit of the woven fiber insulation sleeving used in place of plastic insulation.
Look at the insulators on the points as I (and others) have suggested... they should be on the threaded post with the nut shown in your picture. The assembly order should be, threaded stud, nylon bushing, points spring, condenser ring terminal, ring terminal for wire going to the coil connection (your burned out wire), second nylon bushing, nut. Refer to the picture below. (Sorry the picture is big... I just grabbed it from the net).
Others have listed the wiring above but I will repeat it below. This is for positive earth.
Positive battery lead connected to the engine block.
Negative wire from the battery to the coil SW terminal.
Wire from the coil CB terminal to the plastic lug on the side of the distributor.
From there the current will go through the replacement wire you install in the dizzy to the moving arm of the points via the spring. The path to earth is completed when the points close. The condenser is also connected to the points spring so it is "charging" when the points open.
Again, the damaged wire inside the distributor should not have melted from a short to earth as that's what the points do... they create a short to earth. However, if that wire was partially broken, the current could have burned through what was left of the conductors.
I hope that didn't mislead you.