I'm a bit confused on the reasons why when proving that primitive recursion rule is representable in a PA first order theory one has to utilize ways such as the Gödel function to encode finite sequences of numbers.
It may sound dumb, but could someone try to explain it to me?
From one side we have the mathematical theory of primitive recursive functions, based on a (very limited) initial stock of functions and some rules to build new functions from existing ones.
From the other side, we have formal arithmetic, based on a very limited language: the constant $0$ and the function symbols: $s, +, \times$.
It is very easy to show how to represent (in the technical sense) the initial functions in formal arithmetic.
But what about the rules? Consider the rule of primitive recursion that we have to use to define a function $f(x,n)$ [e.g. the factorial $n!$].
To compute it we have to start from an initial value $f(x,0)=g(x)$ and then a sequence of values $f(x,sy)=h(x,y,f(x,y))$ up to $n$ [in the example above: $0!, 1!, \ldots, n!$ up to $(n+1)!=(n+1)n!$].
And thus the answer to your question: in order to reflect the p.r. definition of a function defined by recursion we have to find a way to express facts about finite sequences of numbers using the resources of formal arithmetic.