Explanation of the demo
A 2900 μF capacitor is charged to 300V then discharged through a 10 ohm 1/4 W carbon film resistor. The resistor doesn't explode so much as it violently flies away. Bellow is a picture of resistor body after it came to rest on the table about 0.5 m away.
Because the time scales involved are "slow" as far as the electrodynamics are concerned, the electric current can be considered constant (spatially uniform) throughout the wire-carbon-wire system. Basically all of the voltage drop occurs across the cylindrical carbon resistor body, so all of the Ohmic heat is generated in the carbon. The temperature of the cylinder rises (rapidly) and the wires to either side heat up due to thermal diffusion from the cylinder. The cylinder eventually fails (whatever that means) and a flaming discharge erupts from the center turning the cylinder body into a rocket. The thrust produced from the flaming discharge rips the cylinder off the now extremely hot wires holding it. The sparks generated are mostly pieces of the molten wire flying away as the cylinder is ripped off the wires and the flash is mostly the flaming discharge that thrusts the cylinder away. The bang is associated with the rapid failure of the ceramic cylinder body.
When watching the high speed videos or looking at the individual frames, it appears that the flash is not emanating from the body as much as it is from an electric arc formed between the two wire leads, take a look at the snap shots from the high speed video bellow. However, the voltage is only 300 V and the distance is several mm so, even at elevated temperatures, there is not nearly enough voltage to produce an electrical arc. In a directional plasma generated through the exploding wire technique (see paper in further reading bellow), gaps in a wire are bridged by electrical arcing, but this is not what is going on here.
High speed video
To see what was going on, I used a high speed camera that could take frames at 7,500 fps at 1 megapixel. If I sacrafice resolution, I could get up to 150,000 fps with this camera. Check out the footage
It appears that you can only use carbon film resistors for this demo. Here is the type of resistor that I used. I also tried metal oxide film and carbon core type resistors. These other types of resistors would fail but, not spectacularly like the carbon film type. The other types would wither fizzle or just short out without any noticeable event.
Not only would I not have known about this demo but, I also would not have done a systematic study of it if it weren't for Kurt Wiesenfeld. Similarly, I would probably have gotten bored and never produced high speed videos of the demo if it weren't for my partner in crime Balachandra Suri who performed many of the experiments with me.