It is important to note that no one supported Einstein's special relativity, on which he based this new but equally controversial and perplexing notion. In short, if special relativity was indeed correct, then in order for both momentum and energy conservation to hold in collisions, the rest energy of a given body when it is not moving (E0) must be equal to the rest mass of this body (m) multiplied by the square of the speed of light. In short, he doubled down, stating that not only should you believe that the special relativity wild idea is true, it necessitates this other massively wild idea.
If the idea that a non-moving body has intrinsic energy is troubling, then I suppose you are equally troubled by a photon, which travels with great speed yet is itself a massless particle. As that photon strikes another body, it actually transfers both energy and momentum to it (that is how solar sails in space work), increasing the body's total energy. The now very slowly moving body retains its original rest energy, but added to that is its newly acquired kinetic energy. Its newly gleaned momentum is achieved due to the change in the photon's momentum.
If we accept that E = mc2, and since it is the most famous equation on the planet (adding the subscript 0 after the E makes it less catchy), I suppose we should, we can deal with the practical consequences, which are as mind-boggling as the concept itself. Let's start with this:
The equation says that if just one kilogram of the rest mass of anything were to be entirely converted into energy (also referred to as "annihilated"), the nuclear reaction would produce 9*1016 Joules; this amount of energy could power a 100-Watt light bulb for 30,000,000 years, or meet the world energy demand for about one hour (using the chemical energy derived from burning coal powers the one light bulb a mere eight hours). To meet the entire world demand for one full year would only require the complete annihilation of less than ten metric tons. This means that if we had the means to safely annihilate entire substances, we could power the globe for decades using only what my father has stashed away in his garage.
It is probably for the best that our species does not yet know how to manipulate nuclear reactions to a greater extent than we already do. After all, look at the devastating impact of atom bombs, which manage to annihilate just a tiny fraction of their mass upon reaction. Similarly, nuclear reactors only begin to tap into the promise of E = mc2, as they convert roughly 0.002% (using enriched uranium) of their mass to viable energy. Over time, should humanity figure a way to release all of the energy within the mass of a given spec of matter in a controlled fashion, it would represent a major shift for society. The notion of an energy crisis would be replaced by a bottomless pit of energy, and ever more need to defend ourselves from ourselves. Sigh.
Why is nuclear power so much more efficient than coal power? It has to do with the nature of the reaction. Burning coal is a chemical reaction known as combustion. A fission reactor houses nuclear reactions, which involves the division of atomic nuclei. The process of nuclear vs chemical is more than 650,000 times as mass efficient. I want to pause for a moment here.
When an engineer optimizes a design, by accomplishing the same thing with 10% less mass, he or she might receive praise. What was effectively done was to reduce the mass by a factor of 1.1. To do so by 650,000 would be like 1.1 to the power of about 140. Therefore, it may be stated, using logic and math, that replacing a coal-burning plant with a nuclear one should receive praise to the power of one hundred forty. Of course, that same logic means that replacing them with solar technology of any type should receive praise to the power of infinity, because those photons are massless (they also travel a distance of one Astronautical unit for free). While we are on the topic, solar energy has no waste to dump anywhere, so it wins, and the sooner we initiate a process to replace all energy infrastructure with solar (combined with large-scale energy on-site storage solutions), the sooner we can begin to think of ourselves as an intelligent species.