What happens when electrons and positrons come into contact with each other?

annihilation and the release of 2 photons

https://en.wikipedia.org/wiki/Electron%E2%80%93pos...

There's an entire wikipedia article on it.

The more interesting question might be why the quarks that make up a proton don't annihilate each other, since they have both positive and negative charges.

But then physics has a lot of trouble answering "why" questions. The tools of science don't really reveal purposes behind the laws we observe, reasons for why the state of nature is such as it is. It observes how things are and forms descriptions of the processes so we can make predictions about future behavior. None of that really fits a "why" questions.

But we can offer some insight into how electrons and protons interact that demonstrate that this just plain can't happen. Why don't electrons crash into the nucleus? Is the first step. If the electron can't actually reach the proton, not annihilation can occur.

We have only ever observed a particle reacting with its anti-particle (e.g. electron and positron), both of which have identical mass, spin, etc. But opposite charge. Only that set of interactions causes the two to combine and form high energy photons. There is no "why", it just is the way the universe's rules are.