Let $X$ be a normed vector space and $M$ a subspace of $X$.
In Folland's text on Real Analysis, he defines an equivalence relation on $X$ as follows: $x \sim y$ iff $x - y \in M$. From this, he denotes the equivalence class of $x \in X$ by $x + M$. What is the intuition behind this equivalence relation? Why is it important in functional analysis?
Later he states a theorem (here he assumes the scalar field is $\mathbb{C}$:
Let $X$ be a normed vector space. If $M$ is a closed subspace of $X$ and $x \in X \setminus M$, there exists $f \in X^*$ such that $f(x) \neq 0$ and $f|M = 0$. In fact, if $\delta = \inf_{y \in M} \|x - y\|$, $f$ can be taken to satisfy $\|f\| = 1$ and $f(x) = \delta$.
In the proof he begins by defining a function $f$ on $M + \mathbb{C}x$. What exactly is the space $M + \mathbb{C}x$? Is it the set of all equivalence classes for $x \in \mathbb{C}$?
To answer the first part of your question, the equivalence relation $x \sim y \Leftrightarrow x - y \in M$ is a preliminary step in defining the quotient space $X/M$. The elements of $X/M$ are precisely the equivalence classes $\{[x]_{\sim} : x \in X\} = \{x + M : x \in X\}$, and you can define addition and scalar multiplication on $X/M$ in the obvious way. The intuition of the equivalence relation is essentially the same as that which underlies quotient spaces in general - you are treating the elements of $M$ collectively as the "zero element" of your newly-defined space. If, furthermore, $M$ is a closed subspace of $X$, then one can define a norm on $X/M$ by $\| [x] \|_{X/M} = \inf\{\|x - m \|_X : m \in M\}$ (if $M$ is not closed, then this will not be a norm in general since you end up with non-zero elements of $X/M$ being mapped to zero by $\| \cdot \|_{X/M}$). And this is how you construct the normed space $(X/M, \|\cdot\|_{X/M})$.
As for the second part of your question, for $x \in X$ (not $x \in \mathbb{C}$ as you presumably unintentionally wrote), you can think of $M + \mathbb{C}x$ as the smallest subspace of $X$ that contains $M \cup \{x\}$. Clearly, if $x \in M$, then $M + \mathbb{C}x = M$. Otherwise, $M + \mathbb{C}x$ is a subspace of $X$ that properly contains the subspace $M$.