Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time step everywhere with a crippling effect on any explicit time-marching method. In [J. Diaz and M. J. Grote, SIAM J. Sci. Comput., 31 (2009), pp. 1985--2014] a leap-frog (LF)-based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time steps in the locally refined region and larger steps elsewhere. Here optimal convergence rates are rigorously proved for the fully discrete LTS-LF method when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities.