The method (I assume) uses Jordan's theorem, which says that an primitive subgroup of $S_n$ with a cycle of prime order (at most $n-2,$ if memory serves) is either $A_n$ or $S_n.$ You rule out $A_n$ by looking at the generators, you show transitivity by randomly generating an $n$-cycle (of which there are a lot, so it does not take long to find one, there are other ways, too), and you show primitivity by finding a permutation which has a $p$ cycle for $n/2 < p < n-1$ (raising it to a power, you just get the $p$-cycle. There are lots of such, so basically generating a few thousand elements will do the trick. Notice that if, after generating the few thousand elements you DO NOT find the sorts of elements you want, the group is almost surely NOT the symmetric group (but the NO answer will be probabilistic, while the YES answer will be deterministic).

For more on this subject, check out my paper with Pemantle and Peres on invariable generation of symmetric groups (the citation thing is not working right now).