What really happens when an electron interacts with a positron?

There are two possible outcomes if an electron and positron collide. The first is the one in which the positron and electron have low kinetic energies. This is when they annihilate, creating 2 gamma radiations (photons). This amount corresponds to the mass rest of the new particle.
The second part of your question is “Why does the mass become energy rather than just combining to make an element?” This can happen, but not in the way that you think. It happens when an electron and a proton combine to form Positronium. (Yes, it sounds science-fiction but bear with me). We know that electrons, and later positrons, have an absolute charge equal to that of the proton. However, the electron one has a negative charge and the positron one has a positive one. Therefore, the electron-positron couple exhibits attraction in the same way as if they were a proton-electron pairing (hydrogen atom).
The simple image of a hydrogenatom shows an electron rotating around a proton, but this is not true for the Positronium. The electron in the hydrogen-atom doesn’t actually revolve around a proton, but rather both revolve around the center mass of the system. Since the proton is heavier than the electron, the center mass can be considered to the proton. Therefore, the electron does not move around it, while the proton does.
The center of mass in the Positronium is equal to the masses of both the electrons and the positrons. This means that we can see the Positronium as an electron following a positron around in a circle, or a positron following an electron depending on your preference.

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