The Mathematical Prediction of Antimatter According to my understanding:
The Dirac Equation, has negative and positive energy solutions. The "Dirac Sea" theory says that these solutions can be interpreted as a "sea" of negative-energy particles, and a "sea" of positive-energy particles, with the same mass, together they form what we perceive as a vacuum. 
What I am confused about, is when there is a "hole" in the negative-energy particle sea, why does the hole have opposite charge to the particle?
 A: This is probably easiest to answer in the solid state case, rather than the particle physics case. In particle physics this is a prediction of antimatter, but in a crystalline solid, electrons are forced by Pauli exclusion to occupy higher and higher energy levels, until they more or less occupy a ball in momentum-space whose surface corresponds to an energy that we call the Fermi energy. In a semiconductor this energy occurs in a “band gap” which does not have any states for electrons to inhabit, between a “valence band” and a “conduction band.”
So, if you put an electric field on the electron sea of a semiconductor, it does what any electric field does: pushes the electrons in the opposite direction.
If those electrons are free electrons in a conduction band, everything makes sense and is straightforward. If they are holes in a valence band, then you have to think backwards a little bit. The electron that fills in the hole is moving opposite the field, so the hole must move along the field. thus, it acts like it has the opposite charge $+1e$ to the electron’s $-1e$.
