Since you asked for a layman's explanation, I'll try to give you one that sticks to the basics (e.g. about what I learned in high school) and hopefully it's not too basic. I'm also going to anthropomorphize the sub-atomic particles since it makes the explanation quicker :)
The basic idea is, the two particles annihilate because they can, and any time something can decay that's probably going to happen. This is largely due to the combination of two physics concepts: minimum energy states, and conservation of quantum states.
The first part says, in essence, that particles "prefer" to be at the lowest mass and energy state they can. Whenever they have "too much" mass or energy, they try to fix it by decaying. This is why high-mass particles, like the ones created in huge particle colliders, decay so quickly. Although mass an energy are equivalent, for various reasons, energy is "preferred" over mass when a particle is deciding what form to take.
The lowest-mass particle we know of is the photon - it has zero mass, and is entirely made of energy. So other particles are pretty quick to give off and/or turn into photons if they can. This is why light bulbs glow: the electrons are giving off some of their own energy as photons so they can be in the lowest energy state possible.
The second idea, though, says that certain quantities about a system cannot ever change. You've probably heard of "conservation of energy" or "conservation of momentum", but lots of physical quantities are conserved, including charge. In order for an electron to decay into something, that thing must otherwise have all of the same conserved quantities, but less mass, and there simply is no such particle. A photon is less massive, but it has no charge, so that isn't an option.
Now, we introduce a positron. The positron on its own has the same problem as the electron: it's charged so it cannot decay into a photon. But it has a positive charge, which is the exact opposite value as the negative electron charge. I'm skipping over lots of details here, including other conserved quantities, but you should get the idea. The key is, other than mass, a positron has exactly the opposite value for each of those conserved quantities as an electron.
When the two collide, all of those other conserved quantities cancel out and become zero. We're left with just an object with the combined mass of the two original particles. There is now no reason not to decay into photons, so that's what happens.