Observe the same process from a frame moving at velocity $v$ relative to the body. Due to relativistic Doppler shift, the total light energy measured in this frame is $L/\sqrt{1-v^2/c^2}$ instead of $L$. This energy difference must come from somewhere — Einstein argued it equals the change in the body's kinetic energy (since the body still moves at velocity $v$ before and after emission, but has become 'lighter').