The point is that you can describe your entire structure within one sentence.
Consider this example: $S=\{<\}$ and $\mathfrak U$ is $\{0,1,2\}$ and $<^\mathfrak U$ is the usual ordering of natural numbers.
We can write: $\begin{align}\varphi:= \exists x\exists y\exists z&\Big(x\neq y\land x\neq z\land y\neq z \land\\ &\forall a(a=x\lor a=y\lor a=z)\land\\ & x
This tells us there are exactly three different elements, and how they are ordered. Every structure in which $\varphi$ is true has three elements and they are ordered as such, we can simply write the isomorphism as $0\mapsto x, 1\mapsto y, 2\mapsto z$.
In the general case, since $S$ has finitely many symbols, and $\mathfrak U$ is finite, we can write an exact description including:
- "There are $n$ different elements in $U$";
- "There are no other elements than those $n$;
- For every function symbol $f$ we can write $f(x)=y$, describing the interpretation of $f$ in $U$;
- For every relation symbol $R$ we can write exactly which $k$-tuples are in $R$ and which are not.
As in the example, it is very simple to write the isomorphism, and prove it is $S$-isomorphism as wanted.