In order to say that a collection of axioms is satisfiable, you must provide a model satisfying them. It certainly appears that standard arithmetic satisfies the Peano axioms, but can you prove it? Can you even prove that it makes sense to talk about "standard arithmetic"? It's not even obvious that infinite sets exist, without an axiom saying they do! Can you show that there isn't a largest integer, without assuming something equivalent to the Peano axioms?
The thing is, if you can provide a finite model satisfying a set of axioms, you can check the axioms directly and thus prove that they're satisfied. For example, to show that the set of sentences $\{a < b, b < c\}$ is consistent, all we have to do is provide the example of three objects $a$, $b$, and $c$, and a relation stating $a < b$ and $b < c$. But Peano Arithmetic has no finite models, so we can't provide anything so directly inspected. In principle, it could be that we've completely misunderstood how the natural numbers work, and that Peano's axioms don't actually hold of them.
Now, most mathematicians take it as assumed that $PA$ is indeed consistent, because it's inconvenient to assume otherwise, but it is definitely possible that a contradiction is provable from $PA$.
As to the distinction "in predicate logic": the Incompleteness Theorem is about predicate logic. Moving to predicate logic doesn't free you from Incompleteness, it puts you right in the crosshairs.
