The paper addresses the propagation of the breather in the quasi-one-dimensional model of locally resonant granular lattice metamaterial on the elastic foundation. The model is presented as a one-dimensional mass-in-mass chain with Hertzian type of nearest-neighbor interaction and linear coupling with the inner elements. In the case of impact excitation on the side of the chain, we demonstrate the formation of the breathers. We consider two types of traveling breathers and detect their existence in a wide range of system parameters. We report that the propagation of the breather is accompanied by energy radiation to the oscillatory tails. Furthermore, we present an asymptotic analysis of the propagation and formulate the criteria for the absence of the energy loss of the breather.