What the section you've quoted is discussing is "axiomatizations characterizing $(N,+,\times,0,1)$ and $(R,+,\times,0,1)$ up to isomorphism." This means axioms which have only a single model, up to isomorphism. The Löwenheim-Skolem theorems tell us a first-order theory with an infinite model has infinite models of every infinite cardinality. This justifies the statement made in the quoted section.
As has been pointed out, this quoted section is not a discussion about finite models for those first-order theories. Clearly no reasonable first-order theory "characterizing" $N$ or $R$ would have a finite model, though one might articulate a first-order theory (e.g. commutative semi-rings) which has both finite models as well as infinite models $(N,+,\times,0,1)$ and $(R,+,\times,0,1)$.
Added: Since the Question was edited to replace "finite model" with "nonstandard model", I'll try to respond to that update. The quoted passage (from Wikipedia) says nothing about nonstandard models explicitly, but the implication is not far from the surface that any first-order theory with "standard" models $\mathbb{N}$ of natural numbers and $\mathbb{R}$ of real numbers, respectively, must have nonstandard models as well (models that are not isomorphic to the standard ones). This is again evident from the Löwenheim-Skolem theorems, since models of different cardinality clearly are not isomorphic.
The formulation "a nonstandard model of natural numbers is not first-order" is suspect because we define "first-order" as a property of a theory (logic+language+axioms), and not in any direct sense as a property of models. Perhaps you mean that it is not possible to "characterize" a nonstandard model of natural numbers with first-order logic, but even so your meaning stands in need of clarification about the kind of characterization sought. If the meaning is "up to isomorphism" (also known as the categorical property of first-order theories), then the Löwenheim-Skolem theorems are sufficient for the purpose for reasons already discussed.