Suppose first that $G$ is Abelian. By definition, $Z(G) = \{ x \in G \mid \forall \, g \in G : x g x^{-1} = g \}$. Since $G$ is a group, the condition for belonging to $Z(G)$ thus obtains the form \begin{equation} \begin{split} x \in Z(G) \quad & \iff \quad \forall \, g \in G : x g x^{-1} = g \\ & \iff \quad \forall \, g \in G : x g = g x \\ \end{split} \end{equation} Since $G$ is Abelian, the predicate expression "$\forall \, g \in G : x g = g x$" is satisfied by every $x \in G$, whence $Z(G) = G$.

The implication in the other direction is established in R755: Group centre is Abelian group. $\square$