Another and probably more natural interpretation of the sentence in the Wikipedia article may be called "localizing an adjunction to an equivalence".

Let $\mathcal C$ and $\mathcal D$ be two categories, and let $F\colon\mathcal C\longrightarrow \mathcal D$ and $G\colon\mathcal D\longrightarrow\mathcal C$ be two functors, with $F$ left adjoint to $G$. Then there are natural transformations $F\circ G\longrightarrow \operatorname{Id}_{\mathcal D}$ and $\operatorname{Id}_{\mathcal C}\longrightarrow G\circ F$, as above.

Denote by $\mathcal S$ the multiplicative class of morphisms in $\mathcal C$ generated by all the morphisms $C\longrightarrow GF(C)$, where $C$ ranges over the objects of $\mathcal C$. Similarly, denote by $\mathcal T$ the multiplicative class of morphisms in $\mathcal D$ generated by all the morphisms $FG(D)\longrightarrow D$, where $D$ ranges over the objects of $\mathcal D$.

Then one can check that the composition $\mathcal C\longrightarrow \mathcal D\longrightarrow \mathcal D[\mathcal T^{-1}]$ of the functor $F$ with the localization functor $\mathcal D\longrightarrow \mathcal D[\mathcal T^{-1}]$ takes all the morphisms from $\mathcal S$ to isomorphisms in $\mathcal D[\mathcal T^{-1}]$. So the functor $F\colon\mathcal C\longrightarrow\mathcal D$ descends to a functor $\overline F\colon\mathcal C[\mathcal S^{-1}]\longrightarrow\mathcal D[\mathcal T^{-1}]$; and similarly the functor $G\colon\mathcal D\longrightarrow\mathcal C$ descends to a functor $\overline G\colon\mathcal D[\mathcal T^{-1}]\longrightarrow\mathcal C[\mathcal S^{-1}]$.

The functors $\overline F$ and $\overline G$ are still adjoint to each other, and this adjunction is an equivalence between the two localized categories:
$$
\overline F\colon\mathcal C[\mathcal S^{-1}]\,\simeq\,\mathcal D[\mathcal T^{-1}]:\!\overline G.
$$

Yet another and perhaps even more natural interpretation of what may be meant by the sentence in Wikipedia also involves passing to localizations $\mathcal C[\mathcal S^{-1}]$ and $\mathcal D[\mathcal T^{-1}]$ of the given two categories $\mathcal C$ and $\mathcal D$ with respect to some natural multiplicative classes of morphisms (often called the classes of *weak equivalences*) $\mathcal S\subset\mathcal C$ and $\mathcal T\subset\mathcal D$.

But, rather than hoping that the functors $F$ and $G$ would simply descend to functors between the localized categories, one *derives* them in some way, producing a left derived functor
$$
\mathbb LF\colon\mathcal C[\mathcal S^{-1}]\longrightarrow\mathcal D[\mathcal T^{-1}]
$$
and a right derived functor
$$
\mathbb RG\colon\mathcal D[\mathcal T^{-1}]\longrightarrow\mathcal C[\mathcal T^{-1}].
$$
Then the functor $\mathbb LF$ is usually left adjoint to the functor $\mathbb RG$, and under certain assumptions they are even adjoint equivalences.

I would not go into further details on derived functors etc. in this answer, but rather suggest some keywords or a key sentence which you could look up: *a Quillen equivalence between two model categories induces an equivalence between their homotopy categories*.