Understanding definition of epsilon closure and finite automata states
Your textbook is being very formal, probably since it aims at using set equations later on. However, $\epsilon$-closure is a very simple concept. The $\epsilon$-closure of a set $M$ of states is the collection of states that can be reached from a state in $M$ by taking any number of $\epsilon$-transitions (possibly zero).

The textbook uses a different (but equivalent) definition: it defines the $\epsilon$-closure of $M$ to be the smallest set of states that contains $M$ and that for every state $s$ that it contains, contains all states reachable from $s$ by a single $\epsilon$-transition.

Regarding the second quote, there is some context missing. Here is one possible context which would make sense. For an NFA without $\epsilon$-transitions, we define $\delta(q,a)$ to be the set of states that can be reached from $q$ by taking a transition labeled $a$. When we have $\epsilon$-transitions, there are several different possible definitions of the transition function. One possibility is to define $\delta(q,a)$ to be the set of states that can be reached from $q$ by taking a transition labeled $a$ and then an arbitrary number of transitions labeled $\epsilon$ (possibly none).