Electrical engineer checking in here.
The cigarette lighter jumpers don’t supply anywhere close to enough current to start the car. They aren’t designed for that. What they do is slowly charge the dead car’s battery from the live car’s battery and alternator. Once the dead car’s battery has enough juice, you use it to start the car.
This has the advantage of being safer (you won’t blow stuff up if you connect it wrong or short something and it’s harder to connect it wrong anyway), but it has the disadvantage that it’s slow since you have to wait for the other battery to charge. It also won’t work at all if the dead car’s battery is completely shot and won’t hold enough of a charge to start the car.
You don’t have to worry about it blowing fuses or burning up the wiring because the thin wire is high enough in resistance to prevent a dangerous amount of current flow. It will let enough current flow to charge the other battery (slowly) but it won’t let enough current flow to be dangerous.
As to the OP, chances are if nothing is damaged now that you’ll be fine. You may have overstressed your alternator a bit, and there is a small chance that you may still see some damage later, but most likely since everything is ok now it will stay that way. Since the other car was dead, it probably wasn’t able to get a negative voltage onto your car’s electronics, which really would have damaged things.
The other car may not have fared so well. Since it was the dead car, it was much more likely to receive a reverse voltage to all of its electronics. As previous posters said, things like the engine computer, alternator, and other components including the wiring may have been toasted. The best case would be that they need a new battery. The worst case is that you are talking about several thousand dollars worth of damage.
Joe Frickin Friday’s instructions for how to make the battery connections are good. Technically you don’t need to make both negative connections away from the battery, but you absolutely want to make sure that you make the last connection to the chassis and not to the battery. Batteries tend to explode for two reasons. The first is that they produce hydrogen while being charged, and that hydrogen can explode due to a spark. The second reason is, if you’ve got a really dead battery or you’ve hooked them up backward, you can get an awful lot of current flowing (enough to melt the insulation off of pretty thick wires, as the OP discovered the hard way), and this excessive current can cause the electrolyte in the battery to boil with the result that the battery explodes. Either way, if you are close to the battery at the time you can get battery acid and bits of plastic and metal sprayed up into your face and eyes, which is just no fun at all.