The definition of :
$F : ω → ℘(\mathcal W^{<ω})$
must start for $n = 0$ with all the finite strings starting with a symbol of arity $0$.
Buy symbols of arity $0$ are symbols without "argument-places" to be filled. Thus the corresponding expressions will be strings of $lenght = 1$ formed by the symbol itself.
I.e. they are the expressions "made of" variables : $x, y, \ldots$ and (if any) individual constants : $c_1, c_2, \ldots$.
The examples in Kunen's book regards terms of first-order arithmetic. If so, we have only one constant : $0$.
Then we "move on" with symbols of arity $1$. This is possible in the language of arithmetic with the (unary) function symbol for the successor function : $S$. In this case we get, e.g. : $Sx, Sy, S0, \ldots$.
Then we have arity $2$, like the (binary) function symbols for sum : $+$ and product : $\times$, generating the terms : $x+y, x \times y, x +0, x +S0, \ldots$.
And so on ...
The easiest case is propositional logic.
You can see Herbert Enderton, A Mathematical Introduction to Logic (2ed - 2001).
We have the set $\mathcal W_0 = \{p_0, p_1, \ldots \}$ of sentential letters.
We hev the propositional connectives : the symbol of arity $1$ : $\lnot$ for negation and the symbols of arity $2$ : $\lor, \land, \rightarrow, \leftrightarrow$.
In this language only $\mathcal W_0, \mathcal W_1$ and $\mathcal W_2$ are not-empty (of course, they are disjoint).
See page 17 for the :
formula-building operations (on expressions) defined by the equations
$\mathcal E_¬(\alpha) = (¬ \alpha)$
$\mathcal E_∨(α,β) = (α \lor β)$,
[...];
and page 32 for the corresponding ones for Polish Notation :
$\mathcal D_¬(α) = ¬α, \quad \mathcal D_∨(α,β) = ∨αβ$, [...].
See [page 30] the description of the Parsing Algorithm :
a procedure that, given an expression, both will determine whether or not the expression is a legal wff, and if it is, will construct the tree showing how it was built up from sentence symbols by application of the formula-building operations. Moreover, we will see that this tree is uniquely determined by the wff.
Finally, see all Section 1.4 : Induction and Recursion [page 34-on] for the Recursion Theorem, which is used to prove the :
UNIQUE READABILITY THEOREM : The five formula-building operations, when restricted to the set of wffs,
(a) Have ranges that are disjoint from each other and from the set of sentence symbols, and
(b) Are one-to-one.
In other words, the set of wffs is freely generated from the set of sentence symbols by the five operations.
Added
The proof is based on the definition by recursion technique :
For any set $A$, any $a_0 \in A$ and any function $G : \omega \times A \to A$, there exists a unique function $\mathcal F : \omega \to A$ such that :
$\mathcal F(0) = a_0$ and for all $n < \omega, \mathcal F(n + 1) = G(F(n), n)$.
We apply it to the definition of the sequence : $n \mapsto \mathsf {Sent_n}$, where :
(i) $\mathsf {Sent_0} = \mathcal W_0$ := the set of sentential letters;
(ii) for each $n < \omega, \mathsf {Sent_{n+1}} := \mathsf {Sent_n} \cup \{ \mathcal D_¬(\varphi) : \varphi \in \mathsf {Sent_n} \} \cup \{ \mathcal D_∨(\varphi, \psi) : \varphi, \psi \in {Sent_n} \}$.
In this definition, the set $A$ is the set $℘(\mathcal W^{<ω})$ of all sets of expressions (of the language), $a_0$ is $\mathcal W_0 \in ℘(\mathcal W^{<ω})$ and $G$ is the function which describes $\mathsf {Sent_{n+1}}$ in terms of $\mathsf {Sent_n}$.
