How does matter turn to energy at the atomic level? When matter is converted to energy by means of $E=mc^2$, it produces quite a lot of "energy". 
What I am having trouble understanding is exactly how the matter is transformed to energy at the atomic level. Do the atoms gain something or lose something in their internal structure? Do they just vibrate at different frequencies when the conversion occurs?
Edit: This is a duplicate of the question, pardon me. Sorry. 
 A: Let us get down to basics, to convert matter to energy the special relativity algebra has to be used. This describes elementary and complex particles by a four vector, whose "length" is the invariant mass of the system described, invariant to Lorenz transformations.


The length of this 4-vector is the rest energy of the particle. The invariance is associated with the fact that the rest mass is the same in any inertial frame of reference. 

The $M$ in the famous $E=Mc^2$  coincides with the invariant mass only in the rest frame of the particle/system, because this $M$ is a function of velocity and is called the relativistic mass and has nothing to do with the energy budget of particles, except at the rest frame of the system.
The fact that energy can be extracted from particles and systems with an invariant mass depends on the quantum mechanical nature of atoms. Atoms are composed out of electrons trapped in the electric potential well of the nucleus, in stable orbitals, , but the energies stored are of order of keV, not really exploitable, also because the orbitals are stable. 
A lot of energy can exist in a nucleus , order of MeV, where neutrons, protons, are bound by the strong force in potential wells, and also where there also exist instabilities that can be exploited, by forcing changes in nuclear structure, i.e. the type and number of nucleons.
This is the binding energy curve 

for the nuclei. It gives for each known nucleus the average binding energy per nucleon, in the collective strong potential well. The fact that one can extract energy from transitions is based on this curve.

Mass defect is defined as the difference between the mass of a nucleus, and the sum of the masses of the nucleons of which it is composed. The mass defect is determined by calculating three quantities. These are: the actual mass of the nucleus, the composition of the nucleus (number of protons and of neutrons), and the masses of a proton and of a neutron. This is then followed by converting the mass defect into energy. This quantity is the nuclear binding energy, however it must be expressed as energy per mole of atoms or as energy per nucleon.

Nuclei where the nucleons, protons and neutrons, are less bound , if fused will give off energy order of MeV. Heavy nuclei, like uranium, broken into pieces with less binding energy will give off energy again order of MeV.
One more link for fusion.
A: 
When matter is converted to energy by means of E=MC2, it produces
  quite a lot of "energy". What I am having trouble understanding is
  exactly how the matter is transformed to energy at the atomic level

If I am reading the question correctly, and you're asking what I think you are asking, it contains a basic misunderstanding, one that is very common due to many craptastic explanations. So maybe this is a slightly less craptastic?
Energy is not a "thing". There is no such thing as "pure energy", no matter what Spock says. Energy is a quality, a property of a thing. A another would be color - can you hold "a yellow" in your hand? No. You can hold a yellow flower in your hand, or a yellow candy, but "a yellow" isn't a thing, it's a quality of a thing. Does that make sense?
So now I'll use an analogy. If you have a chair and you want to get rid of it, you put it up on Craigslist and sell it. Now you have $10. So you go and buy ice cream for you and the friends.
Now would you say that you converted a chair into ice cream? No, it's obvious that this has not happened, the chair and ice cream are two very different things. But they did both cost $10.
And that's where energy comes in. Energy is like the money of the physics world; you can use it to tell you what you have to work with - you're not going to sell the chair and buy a Porsche, and likewise, you're not going to take a low-energy electron and make an atom.
So what E=mc^2 is saying is "there this much potential energy in this system, so whatever you try to make out of it, it can't have more than that".
That's all it says.
Now you're likely aware that this sort of transmutation actually does occur in nature. But it's not "because energy". Each of these changes has a basic physical mechanism that it uses.
For instance, you may have heard that a gamma can turn into an electron and positron. For that to occur, you can say that the gamma must have enough energy to create these two particles, both with the mass of an electron. They mass .511 MeV, so that means the gamma has to be 1.2 MeV or higher. But that doesn't tell you how it happens, only that it can.
So why does this occur? Well that's a lot more complex, of course, and generally requires a nearby nucleus as well. But at no time is there any sort of "pure energy" involved, you start with one sort of particle and end up with another sort of particle. We historically called some of these matter and the others "not", but over time that distinction has blurred. This process is entirely different than the one in fission energy, or fusion, or even burning a log, all of which convert one set of particles to another with different amounts of energy moving about along the way.
