What is the Idea of a Relative Open Set?
As you said, a subset $U$ of $\mathbb{R}$ is open, if for every point $x$ of $U$, there is some $\epsilon>0$ such that $N_\epsilon(x) \subset U$, which means "every point of $\mathbb{R}$ that is closer to $x$ than $\epsilon$ lies in $U$".
Now we want to talk more generally about open subsets of $E$, where $E \subset \mathbb{R}$, but using the same distance function; i.e. keeping as much the same as possible.
So we have $U \subset E$ and we want to define what it means to be open in $E$. The easiest way is to say: for every $x \in U$ there is some $\epsilon > 0$ such that "every point of $E$ that is closer to $x$ than $\epsilon$ lies in $U$". The statement between quotes is just $E \cap N_\epsilon(x) \subset U$, as your text's definition. It makes no sense to consider points not in $E$; we consider $E$ to be the new "universe of discourse", as we only talk about its subsets, no longer about all subsets of $\mathbb{R}$.
A relative open set is essentially the restriction of an open set to a subset. For example, open sets in $\mathbb R$ are such that each point in the set is contained in an open interval or "bubble" as you say. Given this idea of openness, how would we define an open subset of the closed interval $[0,1]$? It seems clear that something like $(\frac{1}{4}, \frac{3}{4})$ should be open but what about $[0,\frac{1}{2})$? This set is not open in $\mathbb R$ because any open interval around $0$ contains a negative number which is not in $[0,\frac{1}{2})$. However, there are no negative numbers in $[0,1]$ so this shouldn't preclude this set from being open relative to $[0,1]$. From inside $[0,1]$, $[0,\frac{1}{2})$ looks like an open neighborhood or your "bubble".
The idea is that you are giving the subset $E$ of $\mathbb{R}$ a topology. A topology on a set is just a collection of subsets $\mathcal{T}$ that satisfy the following three conditions:
$\emptyset$ and $E$ are in $\mathcal{T}$.
$\mathcal{T}$ is closed under finite intersections (i.e. if $U_1,\ldots U_n$ are in $\mathcal{T}$, then so is $U_1\cap\cdots\cap U_n$) .
$\mathcal{T}$ is closed under arbitrary unions.
call the elements of $\mathcal{T}$ open sets of $E$. In $\mathbb{R}$ you know what sets are open. When you take a set together with a topology (called a topological space), you can determine what it means to be open in a space that is not something comfortable like $\mathbb{R}$.
In your example, you are declaring which subsets of $E$ are open. In fact, the topology you are giving $E$ in this context is what is called the subspace topology. You defined what it means for a set to be relatively open in $E$. A nice exercise is to show that the collection of relatively open subsets of $E$ satisfy the properties above.