I am a novice here, but have you looked at Lazarsfeld's chapter "Lectures on linear series" in the book Complex algebraic geometry, IAS Park City math series vol 3?  There he deduces it from Reider's theorem which he deduces from Bogomolov's instability theorem, whose proof he also sketches.  This is a rewrite of some parts of his chapter on application of vector bundles techniques in the book Lectures on Riemann surfaces from Trieste, where however he says his argument has an error.  I am assuming the result you want is that when K is nef then some positive multiple of K is free, (in fact 4K).