The F-111 has, like many aircraft, a fuel dump port so it can get rid of a lot of heavy fuel rather quickly. Most aircraft have the dump ports on the wings, the F-111 designers put it in the tail between the engines. End result is if you dump fuel and briefly light the afterburner you will ignite the liquid fuel in your wake creating a rather spectacular trail of fire.
Aerodynamically it's completely useless, financially it's very expensive, you'd only do it in combat if you were suicidal as it greatly increases your IR signature right where you don't want it, but at an airshow it does look totally badass, especially as the afterburner is adding a lot of noise at the same time.
I would expect any pilot who does a dump-and-burn outside of an airshow (or an emergency) will have his own tail set on fire by the air wing's commanding officer.
The SR-71 had a similar issue - it leaked like a sieve on the ground. Rotating for takeoff would occasionally set fire to the JP-7 on the runway leaving a burning trail behind it much like the Road Runner.
Reply to the video:
watch the takeoff section carefully. You can clearly see the afterburners come on before the fuel dump starts.
Reply to about half the comments:
Highly unlikely they designed it to do this. The F-111 was one of the earlier swing-wing planes, that meant that you can't put much on the wings. And as the bigger tank is in the fuselage anyway, someone would have looked at the space between the engines and thought it was just right for a dump pipe. The flame effect was probably discovered later.