In Introduction to Metamathematics (Kleene) there are a few theorems pertaining to predicates of the the form $$(Ex)R(a_1,...a_n,x) \,\,\,\,\,\,(x)(Ey)R(a_1,...a_n,x,y)\,\,\,\,\,\,(Ex)(y)(Ez)R(a_1,...a_n,x,y,z)\,\,\,\,\,\,...$$
$$(x)R(a_1,...a_n,x) \,\,\,\,\,\,(Ex)(y)R(a_1,...a_n,x,y)\,\,\,\,\,\,(x)(Ey)(z)R(a_1,...a_n,x,y,z)\,\,\,\,\,\,...$$
where $R$ is general recursive.
e.g.
Theorem 35 ($\S72$): The satisfying predicates $P_1,...,P_s$ for F in Theorem 34 [Gödel's Completeness Theorem of pure predicate calculus] can be chosen so that $P_j(a_1,...,a_{n_j}) \equiv (Ex)(y)R_j(a_1,...,a_{n_j},x,y)\equiv (x)(Ey)S_j(a_1,...,a_{n_j},x,y)$ where $R_j$ and $S_j$ are primitive recursive ($j = 1,...,s$).
i.e. For a formula F that is irrefutable in the predicate calculus, there is a satisfying model in the domain of natural numbers such that the predicate letters $P_1,...P_s$ of this model are of the form above.
My question is: Why do we care about theorems like this? Why are number theoretic predicates of this form important?
I would understand the significance of this theorem if we could choose a model so that the predicate letters $P_1,...P_s$ were just general recursive (not $(Ex)(y)R$ where $R$ is general recursive). This way we would could find a model where the predicates are effectively decidable. However, predicates of the form above are not necessarily effectively decidable.
This is the arithmetic hierarchy, and the relevant notation ($\Pi^0_n, \Sigma^0_n,\Delta^0_n$) is quite snappy so I'll use it.
There is a tight connection between syntactic complexity and computational complexity. Specifically:
A set $X\subseteq\mathbb{N}$ has a $\Sigma^0_n$ definition if and only if $X$ is many-one reducible to ${\bf 0^{(n)}}$ (for $\Pi^0_n$, we get many-one reducibility to the complement of ${\bf 0^{(n)}}$).
A set $X\subseteq\mathbb{N}$ has a $\Delta^0_n$ definition (that is, $X$ has a $\Pi^0_n$ definition and $X$ also has a $\Sigma^0_n$ definition) if and only if $X$ is Turing-reducible to ${\bf 0^{(n-1)}}$.
(Here the "${\bf a^{(n)}}$" notation denotes the $n$th Turing jump of the degree ${\bf a}$ - and I'm taking ${\bf 0^{(0)}}$ to just be ${\bf 0}$, the degree of the computable sets, for simplicity.) Moreover, these facts relativize - e.g. replacing $\Sigma^0_n$ with $\Sigma^0_n$ relative to $A$ and replacing ${\bf 0^{(n)}}$ with $deg(A){\bf ^{(n)}}$. We can add more characterizations, and in particular we have Shoenfield's limit lemma and its extensions.
In particular, results of the type you mention provide computational upper bounds.