Is the union of an arbitrary collection of topological spaces a topological space?

There is a distinction to be made here regarding taking unions.
First, it is the definition of a topology that if $X$ is a topological space, then any arbitrary subset $U \subseteq X$ also has a topology on it called the subspace topology which is just $\mathcal{T}_U = \{A\cap U \; | \; A\subseteq X \; \text{is open}\}$. By definition, we are allowed to take an arbitrary union of sets in $X$ and this still remains a subset of $X$ imbued with the subspace topology.

However, a more advanced concept is when you take the union of distinct topological spaces. For instance, if $\{Y_\alpha \; | \; \alpha \in I\}$ is an arbitrary collection of topological spaces, then we can define the disjoint union topology on these spaces to be the union:

$$ Y \;\; =\;\; \bigcup_{\alpha \in I} Y_\alpha $$

where $S \subseteq Y$ is open if $S\cap Y_\alpha$ is open in $Y_\alpha$ for all $\alpha \in I$.

"...collection of connected subspaces of $X$..." means that these have the topology inherited from $X$, i.e., the subspace topology, as does their union.

Munkres is referring to the subspace topology, i.e. given a topological space $(X,\tau)$ and a subset $S\subseteq X$, the subspace topology on $S$ is $\tau_S=\{A\cap S\,:\, A\in\tau\}$.