[Solved]: If $L = L(M)$ then $L$ is a subset of $L(M)$ and $L(M)$ is a subset of $L$

This is just a set-theoretical equality. Two sets are equal precisely when they are included in one another: $A = B$ iff $Asubseteq B$ and $Bsubseteq A$. The notation is applied in the context of languages of automata. In order to show an automaton $M$ defines language $L$, you have to show it accepts all strings from $L$, and no more than those strings. Basic, when doing automata and language theory, I am afraid.