Here 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 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 $μ_\ell$.