Sigma-algebra associated to a stopping time.

Question

I was interested to understand the interpretation of the sigma field associated to a stopping time.
Let $(X_n)_{n \ge 1}$ be a stochastic process and $(F_n)_{n \ge 1}$ its corresponding filtration.
Let $T$ be a stopping time. the associated sigma-algebra is $F_T = \{A \vert A\cap \{T=n\} \in F_n\}$.
I read about this in many many many but I still can't really understand the interpretation. Is it as following? of $F_T$ is as following : if the stopping time has occured at time $n$, i.e. $\{T=n \}$ is realized, then $A \in F_T \iff A \in F_n$. But I can't see step by step why we have this from the definition...