If X is second-countable, then X is Lindelöf.

Let the base be $\{B_n: n \in \mathbb{N}\}$. Let $\mathcal{A}$ be an open cover of $X$.

Then as others already said: define $J \subseteq \mathbb{N}$ by $n \in J$ iff $\exists A \in \mathcal{A}: B_n \subseteq A$.

Then $J$ is a countable set (as a subset of $\mathbb{N}$) and for every $n \in J$ the set $\mathcal{A}_n = \{A \in \mathcal{A}: B_n \subseteq A\}$ is by definition of $J$ a non-empty subcollection of $\mathcal{A}$. (your doubt is that it can have more than one element).

The Axiom of Countable Choice says that there is a function $f :J \rightarrow \cup_n \mathcal{A}_n$ such that $f(n) \in \mathcal{A}_n$ for all $n \in J$.

Then $\mathcal{A}' = \{f(n): n \in J \}$ is as required:

Suppose $x \in X$, then $\exists A_x: x \in A_x \in \mathcal{A}$, as we have a cover. So $x \in B_{n_x} \subseteq A_x$, for some $B_{n_x}$ as $A_x$ is open and the $B_n$ form a base. Then by definition, $n_x \in J$, so $f(n_x) \in \mathcal{A}_{n_x}$, so $x \in B_{n_x} \subseteq f(n_x) \in \mathcal{A}'$, so $\mathcal{A}'$ is a cover of $X$, indexed by the countable set $J$.

Addressing your first doubt, the value of $n$ ranges over the set $J$ of all $n$ such that $B_n \subset A$ for some $A \in \mathcal{A}$. Since the list of $B_n$ is a countable enumeration of the basis elements, $n$ is always a positive integer. Some integers $n$ may not be in $J$, if there is no element of $\mathcal{A}$ containing the basis element $B_n$. But in any case the set $J$ is a subset of the positive integers.

As for your second doubt, what is happening is that for every $B_n$, we choose one $A \in \mathcal{A}$ containing it (if possible, as per the first point), and label it $A_n$. By definition of $J$ this means $n \in J$. We have a surjection from $\mathbb{Z}^+$ to $J$, so $J$ must be countable. Alternately, we have a surjection from the set of all $B_n$ to the set of all $A_n$. Since the former set is countable, so must the latter. Multiple different $A_i$ may contain the same basis element $B_n$, but $B_n$ will have been associated to only one of them . The others will be chosen for different basis elements.

EDIT:

I will add more detail about the construction of the set $\mathcal{A}'$.

First, for every $n \in \mathbb{Z}^+$, define the set $\mathcal{S_n} = \{A \in \mathcal{A}: B_n \subset A\}$. Then define the set $J = \{n \in \mathbb{Z}^+: \mathcal{S_n} \neq \emptyset\}$. Now, we have the countable collection $\mathcal{S} = \{\mathcal{S_n}: n \in J\}$ consisting of nonempty sets.

By the axiom of (countable) choice, there exists a choice function, $f: J \rightarrow \bigcup\limits_{n \in J}\mathcal{S_n}$ such that $f(n) \in \mathcal{S_n}$ for every $n \in J$.

Then define $A_n = f(n)$. Since $A_n \in \mathcal{S_n}$, we have that $B_n \subset A_n$ (by definition of $\mathcal{S_n}$). This is what is meant by "choosing" an element $A_n$ containing $B_n$.

We then have the set $\mathcal{A}' = \{A_n: n \in J\}$.

When I said that "$B_n$ is associated to one of the $A_i$", that was really just a vague way of saying that there is a function $g$ from the set $\{B_n: n \in J\}$ to the set $\mathcal{A}'$. This function would just be defined by $g(B_n) = f(n) = A_n$.

By definition, a function can map $B_n$ to only one value. So only one $A_n$ is "associated" to a given $B_n$, even if there might be multiple $A_i$ containing $B_n$.

But since every $A \in \mathcal{A}'$ is (by definition) equal to $f(i)$ for some $i \in J$, this means $A = f(i) = g(B_i)$ and $g$ is surjective. So every $A$ is "associated" to some $B_i$.

Starting with base $\mathcal B=\{B_n\mid n=1,2,\dots\}$ we arrive at $\mathcal A':=\{A_n\mid n\in J\}\subseteq\mathcal A$.

Here $J\subseteq\{1,2,\cdots\}$ with: $$n\in J\iff B_n\subseteq A\text{ for some }A\in\mathcal A$$ One of sets of $\mathcal A$ that cover $B_n$ is simply picked out and labeled as $A_n$.

It is evident that $J$ as a subset of $\{1,2,\dots\}$ is countable and that consequently $\mathcal A'$ is countable.

Does this makes things more clear to you?