Dear Robert, a reference is Lazarsfeld, Positivity in Algebraic Geometry I, Corollary 2.1.38. Note that the statement there concerns the Iitaka dimension of any line bundle, not just the canonical bundle.
The basic idea of the proof is as follows: given a line bundle $L$ on $X$, one can find a birational morphism $f: X' \rightarrow X$ such that $X'$ fibres over another variety $Y$ of dimension $\kappa_1(L)$. One then compares the numbers $h^0(X,nL)=h^0(X',f^\ast (nL))$ with the number of sections of powers of some ample line bundles on $Y$, which one can compute with Riemann--Roch, to get the desired estimates.