In the remark on the bottom of page 5 of this paper, the author states that

It is well-known fact that the algebra of a Divisor $D$ with $\kappa (X,D) \leq 1$ is finitely generated over $k$. In fact, by the Iitaka fibration, there exists a variety $Y$ of dimension $\kappa(X, D)$, and divisor $D'$ on $Y$ such that $R(X,D) \cong R(Y,D')$.

Assume $X$ to be a complete normal algebraic variety.

Can anyone give a reference for this well-known fact, or elucidate the "$\cong$" in his brief explanation? Of particular interest to me is the case with $X$ a surface and $\kappa(D)=1$.

strictly, is false. There areHalphen surfaces, cf. the excellent answer of Polizzi and comment of Loughran to the following MO question: mathoverflow.net/questions/283609/… In this case, the divisor class $D=-K_X$ has the property that for some integer $m>0$, the subring $R(X,mD)$ is isomorphic to $R(\mathbb{P}^1,\mathcal{O}(1))$. Since $m$ is strictly larger than $1$, $R(X,D)$ is not $\cong R(Y,D')$. $\endgroup$ – Jason Starr Dec 4 '17 at 11:01