It is shown that the geometry of locally homogeneous multisymplectic manifolds(that is, smooth manifolds equipped with a closed nondegenerate form of degree$> 1$, which is locally homogeneous of degree $k$ with respect toa local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend thefamily of classical geometries possessing a similar property: symplectic,volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra ofinfinitesimal automorphisms, and on the study of the local properties ofHamiltonian vector fields on locally multisymplectic manifolds.In particular it is proved that thegroup of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.