I think the usual interpretation is this (see S. Landsburg's comment):

The classical proof that $\mathbb{P}^n$ is proper uses elimination theory: we need to prove that for a ring $R$, $\mathbb{P}^n_R\to Spec(R)$ is closed. Say $R = k[t_1, \ldots, t_k]$, then a closed subset of $\mathbb{P}^n_R$ is given equations in $t_1, \ldots, t_k, x_0, \ldots, x_{n}$, homogeneous in the $x_i$, and finding the image of this subset means eliminating the $x_i$.

Using Chevalley's valuation-theoretic approach ("valuative criterion of properness"), we can replace this rather elaborate argument by a very short and conceptual one. Hence we eliminated elimination theory.

I think the usual interpretation is this (see S. Landsburg's comment):

The classical proof that $\mathbb{P}^n$ is proper uses elimination theory: we need to prove that for a ring $R$, $\mathbb{P}^n_R\to Spec(R)$ is closed. Say $R = k[t_1, \ldots, t_k]$, then a closed subset of $\mathbb{P}^n_R$ is given by equations in $t_1, \ldots, t_k, x_0, \ldots, x_{n}$, homogeneous in the $x_i$, and finding the image of this subset means eliminating the $x_i$.

Using Chevalley's valuation-theoretic approach ("valuative criterion of properness"), we can replace this rather elaborate argument by a very short and conceptual one. Hence we eliminated elimination theory.