To understand the matter-energy conversion you need to understand how quantum field theory describes matter.
Quantum field theory postulates that for every type of particle there is a corresponding quantum field that fills all of spacetime. Particles are described as excitations of these fields. If you add a quantum of energy to a field the energy appears as a new particle. Likewise if you remove a quantum of energy from the field that causes a particle to disappear.
Energy can be transferred between fields. Consider the collision of two protons (actually two quarks) in the LHC. The two quarks have an enormous kinetic energy (14TeV) and that energy can be transferred into other quantum fields where it appears as new particles. That's how new particles, e.g. the Higgs boson, are created in the collision.
The probabilities of the various energy transfers are calculated using quantum field theory. This doesn't allow just anything to happen, for example there are various conservation laws that are always obeyed.
if you were to have, say, a giant capacitor connected to this device, how would this device convert the electricity into matter
At the elementary particle level the concept of energy basically means kinetic energy of particles. If you had some electricity stored on a capacitor you would have to use it to accelerate some particles then let them collide. The collisions could then produce new particles.
Re your recent addition to your question:
Is there a way to do this other than a particle accelerator? Or is that truly the only way?
With a few somewhat esoteric exceptions (e.g. Hawking radiation) matter is produced by converting kinetic energy to new particles in collisions. However these don't have to be in a collider. For example cosmic rays colliding with particles in Earth's atmosphere produce showers of new particles.
Also you ask in a comment:
Some way that would create matter that wouldn't immediately annihilate
To a good approximation the number of particles minus the number of antiparticles is constant. So if you create a particle you must also create a matching antiparticle. The particle and antiparticle are unlikely to annihilate with each other as they probably speed off in different directions, but the antiparticle will quickly annihilate with the other matter around it so the net result is the same.
I say this is an approximation because it is possible to change the net amount of matter in a process known as CP violation. However this is a very small effect and doesn't occuer in most collisions.