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 $X$, after some birational modification (which doesn't change the Iitaka dimension of $L$), $X$ fibres over another variety $Y$ of dimension $\kappa_1(L)$ (in the OP's notation). One then compares the number of global sections of powers of $L$ with the number of global sections of some ample line bundles on $Y$, which one can compute with Riemann--Roch, to get the desired estimates.