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.