There's nothing mysterious or underhand going on here. Though there is an important point in the background, which is simply this:
In a standard formal language (like the language of Peano Arithmetic, but this same applies to any first-order language, or indeed to any sensible formal language) the truth-value of a sentence $A$ will depend only on the interpretation of the vocabulary in the sentence $A$, plus the domain of the quantifiers. What interpretation we give to any additional vocabulary not in $A$ can't affect its truth-value.
Formally, only stuff on the construction tree of a closed wff feeds into determining the semantic evaluation of the wff.
Now, why is this observation relevant to the compactness argument you are worrying about?
- First, generalize: Suppose we have a set of sentences $S$ drawn from a given language $L$. Then the truth-values of those sentences will depend only on the interpretation of the vocabulary in $L$ (plus fixing the domain of the quantifiers).
- Suppose now that we have a set of sentences $S$ drawn from the language $L$, and $S \subset S^+$ where the set of sentences $S^+$ is drawn from the richer language $L^+$ which contains all the vocabulary of $L$ plus some more (maybe extra names, for a start). And suppose we have a model, an interpretation of $L^+$, which makes all the sentences $S^+$ true. Consider the cut-down part of this model that interprets just the vocabulary in $L$ (keeping domains of quantification the same). Then this cut-down model makes all the sentences in $S$ true. Why?
- By hypothesis, the original model, that interpretation of $L^+$, makes all the sentences in $S^+$ true, and since $S \subset S^+$ it in particular makes the sentences in $S$ true. But the interpretation of the bits of vocabulary not in the $S$-sentences, i.e. the bits of vocabulary not in the original language $L$ don't matter by point (2). So the cut-down model interpreting just $L$ will have the same effect, i.e. make the sentences in $S$ true. QED
So now to apply this general observation. If there is a model for the axioms of Peano Arithmetic Plus Some Extra Sentences in Whatever Vocabulary You Fancy then the same model cut down to deal with just the original language of PA will also be a model for the axioms of Peano Arithmetic. That's why is perfectly legitimate tactic to establish that there are non-standard models of Peano Arithmetic by cutting down models of Peano Arithmetic Plus ... (Though as @Mikhail Katz points out in his answer, this route to non-standard models perhaps doesn't give you a "feel" for what the models are like.)