In the Set category, why is every singleton a terminal object?

An arrow in $\mathtt{Set}$ is a map, i.e. a fully defined function. That is if $f: A \to B$ is a arrow in $f$ you can associate a $b\in B$ for any $a \in A$.

Let $X$ be a set and $\{\star\}$ a singleton and let $f : X \to \{\star\}$ be an arrow. Then for $x \in X$, $f(x) \in \{\star\}$. That is, $f(x) = \star$. That completely determines every image of $x$ by $f$ hence it completely determines $f$ to be the constant function to $\star$.

Formally speaking, a map from $A$ to $B$ is a triple $(A,B,f)$ with $f\subseteq A\times B$ that satisfies the following property (called functionnality): $$ \forall a\in A,\left(\exists b\in B,(a,b)\in f\right) \land \left(\forall b,b'\in B, \left((a,b)\in f\land (a,b')\in f\right) \to b=b'\right)$$

For simplicity, we usually denote $f(a)$ for the unique $b$ such that $(a,b)\in f$.

Now if $B$ is a singleton, there is exactly one $f\subseteq A\times B$ that satisfies this property, and this is $A\times B$ itself.