Why is a meteorite impact on the moon visible as white light? You can watch a video of a meteorite impacting the moon at the npr.org website. 

Why does the impact produce white light?

I understand that the impact is like an explosion, but there is no air on the moon to burn material. If it was the result of superheating rock from the impact, then why does it fade so fast. White light emitted by heated rocks should last longer as it takes time to cool.

We can see meteorites breaking up in Earth's atmosphere as a streak of light, but there is no atmosphere on the moon.

So why isn't it just a big black puff of smoke?
The velocity of the impactor of tens of kilometers per second provides enough energy to heat the impactor and parts of the target to several thousands of Kelvins, so that parts are converted to plasma or to vapor, at least.

According to Planck's law the color at these temperatures is white or bluish.

According to the Stefan Boltzman law the total emitted energy is proportional to the forth power of the absolute temperature. Hence material which isn't heated to thousands of degrees, and may glow yellowish or reddish is too dim to be noticible in comparison.

Since most of the heated ejecta are finely distributed, they cool down rapidly by the emitted radiation, and also after contact with the surface.

After cooling there will certainly be some fine dust, which still has to fall back to Moon's surface, but you won't see this on a video from a distance, since it's cooled down and distributed over a wider area.