The link Not Skolem's Paradox - Part 3 includes the following words in a comment :
"By carefully tracing the details of the usual compactness argument and the proof of the completeness theorem, we can see that there must exist a nonstandard model of PA where the carrier set is $\mathbb{N}$ and the addition and multiplication are represented by particular $Σ^0_2$ arithmetical formulas."
My question is - is there a reference where these types of models are reviewed in detail ?
Tennenbaum says there are no recursive models with recursive addition ($\oplus$) or multiplication($\odot$) functions in the metatheory of the form $\mathbb{N}^2$ -> $\mathbb{N}$ that would create such a model. However the above comment suggest, if I have understood it correctly, that there are nonrecursive model solutions for non standard M $\cong$ ($\mathbb{N}$,$\boxplus$,$\boxdot$,<',$n_0$,$n_1$)
If you're just asking why PA has nonstandard models with carrier set $\mathbb{N}$, this is an immediate consequence of the Lowenheim-Skolem theorem: if $\mathcal{M}\models PA$ is nonstandard, let $c\in\mathcal{M}$ be an infinite element (i.e. $\mathcal{M}\models c>S(S(S(...(0))))$ for every finite string of $S$s); then there is a countable elementary $\mathcal{N}\preccurlyeq \mathcal{M}$ with $c\in\mathcal{N}$. $\mathcal{N}$ is nonstandard as well, since $c\in\mathcal{N}$, and since $\mathcal{N}$ is countable we may fix a bijection $f$ from the carrier set $A$ of $\mathcal{N}$ to $\mathbb{N}$, and this provides a nonstandard model of PA with carrier set $\mathbb{N}$ (just "push" the structure of $\mathcal{N}$ through $f$).
The nontrivial feature is a model with carrier set $\mathbb{N}$ whose functions are "not too complicated" (that is, $\Sigma^0_2$). This is a consequence of a much more general fact. Henkinization is effective, if we're given a consistent complete theory with the witness property to start with. So the complexity of building a nonstandard model of PA comes down to the complexity of extending the theory $$PA\cup\{c>S^n(0): n\in\mathbb{N}\}$$ to a consistent complete theory with the witness property. It's not hard to show that any consistent computable theory can be extended to a consistent complete theory with the witness property, effectively in $0'$; so $0'$ computes models of any consistent computable theory, with carrier set $\subseteq\mathbb{N}$.
Now since the sets computable in $0'$ are exactly the $\Delta^0_2$ ones, this means that any consistent computable theory has a model with carrier set $\subseteq\mathbb{N}$ which has complexity $\Delta^0_2$ (even better than $\Sigma^0_2$!). This can be substantively improved: call a degree $d$ "PA" if it computes models of any consistent computable theory. Then there are low PA degrees - that is, PA degrees $d$ satisfying $d'=0'$. For more info on this, see here.