The Peano Axioms are just that: a bunch of statements expressed in the language of first-order logic. And notice that they don't use the $1$ symbol, so you're right: how can we tell that $s(0)$ is supposed to be $1$?
But please note that in a way it is even worse than you think: why should the $0$ logic symbol even be the same as the mathematical $0$?! Again, as far as purely formal logic is concerned, it's all just a bunch of symbols. So for all we know, $0$ is referring to bananas, and $s$ is the function of putting something in a smoothie-maker, which would make $s(0)$ to be a banana-smoothie.
But of course, in creating the Peano Axioms, we didn't aim to be talking about bananas and smoothies. We intended the axioms to be about the natural numbers. Specifically, we use the $0$ symbol to represent the mathematical number $0$, and we used $s$ for the 'successor' function, and since $1$ is the successor of $0$, we can treat $s(0)$ as the number $1$.
So notice that we don't prove that $s(0)$ is $1$, but rather stipulate (define, if you want) that $1$ is the number that comes after $0$. That is, it is our intended interpretation that tells us that $s(0)$ is the same as $1$.
Interestingly, your question as to how we know that the successor of $n$ is $n+1$ actually is not a mere definition. Indeed, your question amounts to proving that $s(n) = n + s(0)$ ... and that is something we can do using the Peano Axioms that relate to addition:
$n + s(0) = s(n+0) = s(n)$
Hey, a first theorem we can derive from the Peano Axioms!
P.s. Using the Peano Axioms you can prove that $s(s(0)) \neq s(0)$. So I guess this means that the Peano Axioms can't be talking about banana-smoothies, since if you put a banana-smoothie into a smoothie-maker, you get back that very same banana-smoothie :P