How is this read correctly in maths , and meaning (about: sets)?

The "fix" means "consider some particular $J$" or "for each $J$ define $S$ to be ..." . Your description of $S$ in terms of $J$ is correct.

No need to overthink this.

I would not use the word "fix", since logically speaking what is going on here is that we want to consider any arbitrary given $J ⊆ \{1..n\}$. Much better would be something like "Take any $J ⊆ \{1..n\}$, and let $S = \cdots$. Then . . .". But why do people use this word? It is because in their mind they are thinking that they want to temporarily focus on one specific $J$ and reason about that set together with another set $S$ associated with that particular $J$. So this becomes "let's fix $J$ for now, and define $S$ in terms of this $J$, and ...". Of course, in the end we are not talking about only one constant fixed $J$. This notion can be made clear in Fitch-style natural deduction:

[We want to prove $∀n{∈}ℕ\ ∀J{⊆}\{1..n\}\ ( \ Q(n,J) \ )$.]
Given $n{∈}ℕ$ and $J{⊆}\{1..n\}$:
  [It suffices to prove $Q(n,J)$ in this subcontext (i.e. with $n{∈}ℕ$ and $J{⊆}\{1..n\}$ given to you).]
  [Note that in this subcontext you have no control over $n,J$; they are fixed here.]
  $\vdots$
  $Q(n,J)$.
$∀n{∈}ℕ\ ∀J{⊆}\{1..n\}\ ( \ Q(n,J) \ )$.   [In the conclusion $n,J$ are of course not fixed.]