It is said that transpositions of a finite set $E$ form a conjugacy class in the group $\mathfrak{S}_{E}$, the symmetric group of the set E.
I am not sure what is meant by this.
Let $G$ be a group. I understand that the conjugacy classes are orbits of the operation of $G$ on itself by inner automorphisms. But I don't know how to apply this to the case at hand. What does the corresponding homomorphism of this operation look like? And what is the corresponding equivalence relation?
Transpositions are bijections $\sigma : E\to E$ which fix all but two elements and switch two. The action of G on E by conjugacy is the map $G \to Sym(E)$ defined by $s\mapsto g\sigma g^{-1}$ The orbits of this action are the conjugacy classes of elements in the set E, like you've noted. Since E is finite, note $Sym(E) = S_n$ for some n, so you basically know the answer you just need to understand the conjugacy class of any transposition in $S_n$. You can show that this is indeed the set of all transpositions in $S_n$.
The equivalence relation corresponding to this is that $\sigma $ and $\tau$ are equivalent if they are conjugate. i.e. there exists some $\alpha \in S_n$ such that $\sigma = \alpha \tau \alpha^{-1}$.