Why did Air France flight 447 fail to recover from the stall? I have read the wikipedia entry about the accident but I don't understand why did the stall happen and why the plane couldn't recover from the stall. 

Correct me if I am wrong but I would think a plane would naturally recover from a stall situation by pitching down the noise by itself with no control inputs needed and gaining speed (due to the fall).  The AoA would be back to safe values to generate lift again and the plan would be stabilized.  

I don't get how a plane like that would fall from 35,000 feet with its engines set at 100% thrust.
I‘ll attempt a very brief answer based on the report against your questions:

1) Why did the aircraft stall?
The aircraft stalled because crew pitched the aircraft up beyond the performance limit at that altitude (they pitched into a climb which the available engine thrust could not sustain, bleeding off speed until the aircraft stalled).

2) Why was power not reduced?
I don’t know, but in my view that didn’t play a crucial role, either.

3) Why did the aircraft not pitch down by itself?
Because crew didn’t realise they were stalled and continued to apply „nose-up“ control input. Seemingly inconsistent behaviour of the stall warning system and unexpected parameters on their instruments (i.e. a combination of speed and vertical speed not ever seen in normal flight) contributed to the situation.