Viewing a discrete category with a single object as the full subcategory of $\textbf{Set}$ generated by a singleton set. 
Let us write $\mathbf{1}$ for the discrete category with a single
object $\star$;
$$1:\mathbf{1}\rightarrow\textbf{Set},\,\star\mapsto\{*\}$$ is the
functor which maps the unique object $\star$ of $\mathbf{1}$ to the
singleton $\{*\}$. In other words, we view $\mathbf{1}$ as the full
subcategory of $\textbf{Set}$ generated by a singleton set.

I am not clear about the claim "... we view $\mathbf{1}$ as the full subcategory of $\mathbf{Set}$ generated by a singleton".
Is "$\{*\}$" meant to be "$\{\star\}$"? Otherwise, how can we have $\text{Ob}(\mathbf{1})\subset\text{Ob}(\textbf{Set})$, as required by the definition of subcategory?
 A: Think of this example instead. Obviously, $\mathbb N$ is a subset of $\mathbb Z$, right? Well, under the usual implementation of $\mathbb Z$ as equivalence classes of pairs of natural numbers, $\mathbb N\not\subseteq\mathbb Z$. However, there is an obvious injection $n\mapsto (n,0)$ which we use to identify natural numbers with appropriate integers. After we prove the stuff we need for the constructed $\mathbb Z$, we immediately forget the implementation and equivalence classes and just write $\mathbb N\subseteq \mathbb Z$. It makes sense, right?
It's kind of the same here. Note the wording: "In other words, we view $\bf 1$ as the full subcategory..." (emphasis mine) which means that we identify $\bf 1$ with a particular subcategory of $\bf Set$ isomorphic to it and view it as a subcategory when strictly speaking it is not.
A: Not necessarily—in fact, writing $\mathbf{1}'$ for the full subcategory of $\mathbf{Set}$ given by the image of the functor $1 : \mathbf{1} \to \mathbf{Set}$, we have $\mathrm{ob}(\mathbf{1}) = \{ \star \}$ but $\mathrm{ob}(\mathbf{1}') = \{ \{ * \} \}$, so even when $* = \star$, the categories $\mathbf{1}$ and $\mathbf{1}'$ don't have the same set of objects.
There is no need to require $* = \star$, as I'll now elaborate, but replacing the symbol '$\star$' by '$x$' for better readability.
If $x$ is any object at all (mathematical object, that is, not necessarily an object of a category), then the full subcategory of $\mathbf{Set}$ whose unique object is $\{ x \}$ is isomorphic to $\mathbf{1}$, and so $\mathbf{1}$ can be included into $\mathbf{Set}$ in many different ways.
In fact, embeddings $I : \mathbf{1} \hookrightarrow \mathbf{Set}$ with $I(\star) \ne \varnothing$ correspond exactly with singleton sets $\{ x \} \in \mathrm{ob}(\mathbf{Set})$.
