That definition seems to assume that $M$ is a finite set, since the members of a simplicial complex are normally required to be finite subsets of the underlying set. Yes, $\Delta_M$ is the power set of $M$. If $\Gamma\subseteq\Delta_M$, $\Gamma$ is closed under the taking of subsets if and only if $X\subseteq Y\in\Gamma$ implies that $X\in\Gamma$.
More generally, if $\mathscr{A}$ is any collection of sets, we say that $\mathscr{A}$ is closed under the taking of subsets (or simply closed under taking subsets) if every subset of a member of $\mathscr{A}$ is also a member of $\mathscr{A}$. Yet another way to say this is that for each $A\in\mathscr{A}$, $\wp(A)\subset\mathscr{A}$. Unions are not involved here.
If $K$ is a geometric simplicial complex, let $M$ be the set of its vertices; we’ll build an associated abstract simplicial complex $\Gamma$ on $M$. Each $d$-dimensional face of $K$ has $d+1$ vertices; let that set of $d+1$ vertices belong to $\Gamma$. For example, if $K$ consists of a tetrahedron with vertices $v_1,v_2,v_3$, and $v_4$, a triangle with vertices $v_1,v_2$, and $v_5$, and a segment with vertices $v_5$ and $v_6$, $M=\{v_1,v_2,v_3,v_4,v_5,v_6\}$, and
$\begin{align*}\Gamma&=\Big\{\{v_1,v_2,v_3,v_4\},\{v_1,v_2,v_3\},\{v_1,v_2,v_4\},\{v_1,v_3,v_4\},\{v_2,v_3,v_4\},\\ &\quad\;\;\,\{v_1,v_2\},\{v_1,v_3\},\{v_1,v_4\},\{v_2,v_3\},\{v_2,v_4\},\{v_3,v_4\},\{v_1\},\{v_2\},\\ &\quad\;\;\,\{v_3\},\{v_4\},\varnothing,\{v_1,v_2,v_5\},\{v_1,v_5\},\{v_2,v_5\},\{v_5\},\{v_5,v_6\},\{v_6\}\Big\}\;. \end{align*}$
Going in the other direction, I’ll quote Wikipedia:
If $K$ is [a] finite [abstract simplicial complex], then we can describe $|K|$ more simply. Choose an embedding of the vertex set of $K$ as an affinely independent subset of some Euclidean space $\Bbb R^N$ of sufficiently high dimension $N$. Then any face $X \in K$ can be identified with the geometric simplex in $\Bbb R^N$ spanned by the corresponding embedded vertices. Take $|K|$ to be the union of all such simplices.
(Here $|K|$ is the geometric realization of $K$.)