Here's what I gather:

You can define the natural numbers using certain rules (the Peano axioms). These rules make reference to sets, so they are in turn governed by the rules of set theory (ZFC), which admit several models.

What the paper goes on to talk about is that even if you just use the natural numbers--not some weird system--then there will still be some statements whose truth depends on the ambient model of set theory. In other words Model 1 and Model 2 of ZFC can both admit the natural numbers, but they can still disagree about what's true. Certain things will always be true, but others will be true or false depending on which model you use.