How does a fuel tank fire self-extinguish in an A380?
The Australian Transportation Safety Board's Final Report gives a couple of possible reasons that the reaction was not sustained in Appendix D on page 257:


  
  The flash fire initially consumed the oxygen inside the tank; reducing the oxygen concentration level to below the threshold required for sustained combustion of Jet A-1.  
  The general temperature rise associated with the initial combustion was low and insufficient to increase the temperature of the fuel inside the tank to at least the lower flammability limit of Jet A-1. Quenching of the flame by the cool fuel tank walls (evident from the sooting) and the entry of cooling air through the tank perforations may have promoted this.
  


Note that Jet-A requires quite a bit of activation energy to vaporize and burn. If the burning of what fuel did burn didn't produce enough energy to provide activation energy for the surrounding fuel, the fire would not be sustained. The fire was likely very brief as the turbine disc fragment passed through the tank.

From what I've heard (I haven't tried this myself,) you can drop a lit match in a bucket of Jet-A and it won't ignite. Using fuel that doesn't easily burst into flames in a vehicle designed to carry tens of thousands of gallons of fuel at several hundred miles per hour is desirable and intentional for obvious reasons.

To illustrate just how hard it is to burn Jet-A, the fuel tanks in Asiana 214 never ignited. The entire fuselage was burned out, but the fuel tanks never burned.


Remnants of Asiana 214's hull Source: Wikipedia