Understanding an example of forcing In Chapter 14 of Jech's Set Theory, example 14.2 is as follows:

Example 14.2. Let $P$ be the following notion of forcing: the elements of $P$ are finite $0$-$1$ sequences $\langle p(0),\dots,p(n-1)\rangle$ and a condition $p$ is stronger than $q$ if $p$ extends $q$. Clearly, $p$ and $q$ are compatible if and only if $p\subseteq q$ or $q\subseteq p$. Let $M$ be the ground model (note that $(P,<)\in M$), and let $G\subseteq P$ be generic over $M$. Let $f=\bigcup G$. Since $G$ is a filter, $f$ is a function. For every $n\in \omega$, the sets $D_n=\{p\in P:n\in \text{dom}(p)\}$ is dense in $P$, hence it meets $G$, and so $\text{dom}(f)=\omega$.
The $0$-$1$ function $f$ is the characteristic function of a set $A\subseteq \omega$. We claim that the function $f$ (or the set $A$) is not in the ground model. For every $0$-$1$ function $g$ in $M$, let $D_g=\{p\in P:p\not\subseteq g\}$. The set $D_g$ is dense, hence it meets $g$, and it follows that $f\neq g$.
This example describes the simplest way of adjoining a new set of natural numbers to the ground model. A set $A\subseteq \omega$ obtained in this way is called a Cohen generic real. Except in trivial cases, a generic set does not belong to the ground model; see Exercise 14.6.

I understand each individual step, but I don't really get what this is actually doing. It says something about "adjoining a new set of natural numbers to the ground model," but I don't really know what that means or how this process is accomplishing that.
I'm also not totally convinced why a generic set always has to exist (i.e.how we know we can let $G\subseteq P$ be generic over $M$). Is there a simple way to see this?
 A: To get "there is a generic set", you can just build one, assuming we're in a countable model of ZFC. Enumerate the dense subsets of $P$ (there are only countably many of these, because the model of ZFC is countable), letting $D_n$ be the $n$th dense subset of $P$. Then we can inductively construct an increasing sequence $d_0, d_1, \dots$ with $d_i \in D_i$, by density of each $D_i$. Now just let $G := \{ q \in P: (\exists n)(q \le d_n) \}$. $G$ is downward-closed, and for every $p, q \in G$ there is an upper bound in $G$ (namely $d_{\max(m,n)}$ where $p \le d_m$ and $q \le d_n$), and it was constructed to meet each $D_i$ at $d_i$, so it's generic.

Why is it obvious that there are "missing" sets of naturals in the first place? Because the original model was countable, so it can't contain every set of naturals! But the clever bit is in finding a way to extend the original model by adding one of the missing sets without breaking the axioms of ZFC, without doing bizarre things to the notion of truth, and without breaking too many of the statements that were true in the original model.

For "how is this actually adding a set", I refer you to https://math.stackexchange.com/a/1174805/259262 which discusses what a $P$-name is; I find it conceptually easier to unpack everything, because I'm basically a novice, so the rest of this answer is rather hand-wavy and is not backed by as much rigour as I'd like. I've definitely not fully grokked forcing myself. I've also never read Jech, so don't know what approach is taken there and whether this fits into it; if you haven't seen $P$-names yet, then the rest of this answer may not help at all.
The point is that we're using forcing to take an existing model of ZFC (for example) and build a new model via the $P$-name construction: we define the set of $P$-names in a certain cunning way, then we pick some generic set $G$ and interpret every name against that $G$ to obtain the new model. Then we need a whole lot of theorems to show that this is actually a model of ZFC, that it contains the original model, it's countable, the new model is a model of any $\phi$ iff there is a $p \in G$ which forces $\phi$ over the original model, etc. Moreover, as Jech refers to at the end of your snippet, it's usually the case that there are names which don't name anything from the original model (no matter what $G$ is picked to evaluate the name), so in fact this is usually building a genuinely new model with "more things in it": for example, $G$ itself is usually not in the original model but is the result of evaluating the name $\{(p, \check{p}) : p \in P\}$ (where $\check{p}$ is the canonical $P$-name of $p$).
In your case, Jech describes a forcing $P$ and then exhibits a set which is different from every set of naturals from the original model, but which we can name using $P$. (It might help to walk through the construction of one possible $G$ I outlined above, and see exactly why it's different from every existing set, and work out a name for it. But it's late at night so I haven't done this myself.)
