As he points out, one wants to show that a Sylow $p$-subgroup of the unknown $G$ is of index $(p-1)/2$ in its normalizer (which plays the role of Borel subgroup $B$ in a BN-pair of rank 1). Then you are looking at a doubly transitive permutation group on the $p+1$ elements of the coset space $G/B$ (thought of as rational points of a projective line).
I've just looked at the relevant Frobenius paper, being probably the only person in the U.S. with all three volumes of his collected papers on a shelf several feet from my home computer. (Not that I've read them all. It was an extra set owned by Wilhelm Magnus.) The 19 page 1902 paper Uber Gruppen des Grades $p$ oder $p+1$ was published in the Sitzungsberichte ...; it appears as number 66 in the third volume of collected papers containing later work on finite groups and character theory.
The paper is somewhat electic, but the early Satz II affirms for any prime (except 7) the existence of a unique transitive permutation group of degree $p+1$ and order $p(p^2-1)/2$. This requires four pages or so of heavy proof, with reference back to Sylow and others. It's hard to see at a glance how this translates into current terminology, but for example he sees very soon that the group permutes $p+1$ objects which he denotes $\infty, 0, 1, \dots, p-1$. Once all the work is done, it follows as Satz III that for $p>3$ there is a unique simple group of the given order.