An easy way to make E=mc2 understandable to children is to start with a paraphrase of the first law of thermodynamics: "Nothing disappears; everything goes somewhere." Matter, for example, cannot be destroyed, but only turned into energy. E=mc2 is the equation that describes the amount of energy it turns into.
Matter can be thought of as highly organized energy, and energy can be viewed as disorganized matter. Turning one into the other can only be done at a certain ratio, and that ratio is what Einstein's equation provides. Splitting a heavy atom makes two lighter atoms, but their combined weight is not equal to the weight of the original atom; it's slightly lower. The difference in mass was converted to energy at a rate of E(energy released) = m(difference in mass) x c(the speed of light)2.