Gravitational waves originate in much the same way as electromagnetic waves do. If you accelerate an electric charge, like an electron, it will produce an electromagnetic wave. The energy for this wave comes from the kinetic energy of the electron.
In the same way, if you accelerate a mass, it will emit a gravitational wave, and the energy for that wave will come from the masses kinetic energy.
In fact, since the above electron also has a mass, it will emit gravitational waves as well as electromagnetic ones. Gravitational waves however are very very weak and the electromagnetic waves account for the vast majority of the lost kinetic energy.