One reason is Givental's conjecture, which says that in the semisimple case, genus 0 GW invariants determine higher genus GW invariants. See this paper of Teleman, in which the conjecture is proved.
The theory of Frobenius manifolds in general is quite complicated. I guess semisimple Frobenius manifolds form a relatively tractable set of examples. Here are some basic references for the theory of semisimple Frobenius manifolds:
Manin: Frobenius manifolds, quantum cohomology, and moduli spaces
Lee, Pandharipande: Frobenius manifolds, Gromov-Witten theory, and Virasoro constraints (so far unpublished and incomplete; available at Pandharipande's webpage)
Semisimple Frobenius manifolds also arise in singularity theory, when studying for instance isolated hypersurface singularities (see Hertling's book Frobenius manifolds and moduli spaces for singularities; the three references above probably also talk about this), or in the physics terminology "Landau-Ginzburg (B-)models". I don't know whether Frobenius manifolds (in particular non-semisimple ones) arise more generally in singularity theory...? In any case, these Frobenius manifolds coming from singularity theory are supposed to be related to those coming from Gromov-Witten theory via mirror symmetry.*
Another comment: Quantum cohomology of, for example, $\mathbb{P}^n$ is semisimple. Then perhaps this makes quantum cohomology and GW theory of projective varieties more tractable, because of quantum Lefschetz ... but I don't really know anything about this. But very roughly speaking, I think this is the strategy of Givental in his proof of the "mirror conjecture" of Candelas et. al. regarding the genus 0 GW theory of the quintic 3-fold, though I might be wrong.
*Edit: For example, this paper of Etienne Mann seems to prove a mirror theorem relating the quantum cohomology Frobenius manifolds of (weighted) projective spaces and the Frobenius manifolds associated to the mirror Landau-Ginzburg B-models. As Arend mentions, germs of semisimple Frobenius manifolds are specified by a finite set of data, and I think the strategy of Mann's paper is to show that these data coincide for the two Frobenius manifolds.