As the title says.
Let's consider some category $C$ and the contravariant functor $Sub : C^{op} \rightarrow Set$ that for each object gives its set of subobjects.
The introductory book by Leinster says that by the Yoneda lemma, $Sub$ being representable amounts to there being an object $\Omega$ and an element $t:Sub(\Omega)$ which satisfy the definition of subobject classifier, and gives no more explanation.
How is he using the Yoneda lemma exactly here?
A functor $F : C^{\text{op}} \to \mathrm{Set}$ being representable means that there is an object $c$ of $C$ and a natural isomorphism $C(-, c) \to F$. The Yoneda lemma gives a bijection $[C^{\text{op}}, \mathrm{Set}](C(-, c), F) \cong Fc$. Therefore a natural transformation $C(-, c) \to F$ (which is an element of the left hand side of this bijection) is the same thing as an element of $Fc$ (which is the right hand side of this bijection). So a functor $F : C^{\text{op}} \to \mathrm{Set}$ is representable just in case there is an object $c$ of $C$ and an element $x$ of $Fc$ such that the element of $[C^{\text{op}}, \mathrm{Set}](C(-, c), F)$ corresponding to $x$ under the Yoneda lemma is a natural isomorphism.