The idea goes like this. Normal electrons are particles that occupy a positive energy regime of unfilled states (the positive continuum). In the negative energy continuum, all energy states are filled. A high energy photon can come along and knock an electron out of this negative region, and promote it to the positive energy continuum, leaving behind a "hole". This hole will have an effective positive charge,a nd can be thought of as a particle in itself, i.e. the positron. if you apply and electric field, one would see that all electrons move in teh same direction (say to teh right), making it seem that the hole moves to the left, which is teh direction a positively charged particle would move in response to that same field.
So when an electron and a positron come together again, the electron merely fills the hole in the negative energy continuum, and the excess energy is given out again in high energy photons. Thus the electron and positron disappear (or annihilate) and are replaced by photons.
Why we dont experience this negative energy continuum is another matter, and one I dont know the answer to, but it was Diracs vision of the model, and it works well to illustrate the concept. Indeed, the concept is used and works well for semiconductor physics, where both electrons and electron holes contribute to conduction behaviour in these materials (though the holes in this case are not true positrons, but are actually absences of electrons from energy states in the band structures of these materials)
This post was created using 100% recycled electrons