It is not hard to see why. Seal a nuclear reactor and a battery in a box.
Nothing can enter or leave the box.
Now let the reactor put out electric energy and let that energy be put into the battery. As the reactor puts out energy it must lose mass. But no mass can get out of the sealed box. The only other place it could be is in the battery. So the battery gains mass as it gains energy, and the battery loses mass as it puts out its energy.