[Here][1] are some lectures by Charles Weibel.  Early on, they discuss Milnor Conjecture and Bloch-Kato, and they should go through the proof.  My understanding is that there were a bunch of people involved in the proof, though a few were a bit reticent to actually write up their parts of it, and so Weibel drew the short straw and is the one writing it up.

EDIT: Adding a bit, [here's][2] Weibel's 2006 page where he notes the status as of then, and to make sure that this is roughly self-contained, here's the statement:

> For an odd prime $\ell$, and a field $k$ containing $1/\ell$, the Milnor K-theory $K^M_n(k)/\ell$ is isomorphic to the étale cohomology $H^n_{ét}(k,μ_\ell^n)$
of the field $k$ with coefficients in the twists of $μ_\ell$. 


  [1]: http://www.math.rutgers.edu/~weibel/papers-dir/Bloch-Kato.pdf
  [2]: http://www.math.rutgers.edu/~weibel/motivic2006.html