The $^*$ notation does not mean excited in this case, it means "off shell" (i.e. virtual or having the "wrong" mass).
At the second vertex the $Z^0$ is put "on-shell" by the emission of a Higgs (note, however, that it will decay very quickly in any case).
The lepton pair can annihilate directly to the Higgs, but the event is experimentally identical to annihilation to photons or $Z^0$s (because the thing that makes a coupling possible is that both side have compatible quantum numbers, so that (at tree level) all three possibilities decay to very similar end states).
The reaction pictured is experimentally identifiable because the on-shell $Z$ decays to an lepton pair with a mass of 90 GeV and the Higgs decays to a limited choice of end states that are mostly reconstructable and add up to the Higgs mass.
A surprising amount of collider physics is not so much about what can happen as about what can be uniquely shown to have happened.
Finally, I would certainly not describe this reaction as a "decay".