The connected components of a graph partition its vertex set and are its maximal connected subgraphs

Reflexivity in [L1] gives $v\in C_G(v)$, so every class is nonempty and the classes cover $V$.

If $C_G(u)$ and $C_G(v)$ meet at $w$, then $u$ reaches $w$ and $w$ reaches $v$ by symmetry; transitivity gives that $u$ reaches $v$, and another use of symmetry and transitivity shows a vertex reaches $u$ exactly when it reaches $v$. Thus $C_G(u)=C_G(v)$, so distinct classes are disjoint.

If $x,y\in C_G(v)$, then $x$ reaches $v$ and $v$ reaches $y$; concatenating these paths gives a walk from $x$ to $y$, and deleting any repeated segments gives a path all of whose vertices remain reachable from $v$. Hence $G[C_G(v)]$ is connected.

If a connected subgraph $H$ contains $v$, every vertex $x$ of $H$ is joined to $v$ by a path in $H$, hence also in $G$, so $x\in C_G(v)$. Thus $H$ is a subgraph of $G[C_G(v)]$.

Steps 1.1 and 1.2 give the partition assertion, and steps 1.3 and 1.4 say exactly that its induced subgraphs are connected and maximal among connected subgraphs.