That's hard to answer for a few reasons. I'm not a particle physicist, the subject is kind of complicated, and most people start off ill-informed (sorry!).
Antiparticles are not particularly weird and particle-antiparticle interactions are, in particular, not some kind of physical witchcraft. I always have disliked that it's called annihilation. At the subatomic level, particle interactions are common and they generally involve the "creation" and "destruction" of particles. For example, maybe a neutron decays into a proton, an electron, and an electron antineutrino (by way of one of its down quarks changing into an up quark). Particle interactions are all sort of a shuffling of energy between the different flavors of bundles of energy we call particles. Lots of different physical quantities, like charge, are conserved, limiting what interactions can happen.
In the interest of simplicity, a lot of what I'll say next is slightly wrong.
Antiparticles aren't particularly weird. Particles all have a set of physical properties. It turns out that for each particle, there is another particle that is basically exactly the same, except all these physical properties are opposite. So an electron has charge -1 and an antielectron (positron) has charge +1. In fact, if you look at a legal particle interaction and replace all of the particles with their antiparticles, it's still a legal particle interaction.
An implication of this is that if a particle and its antiparticle interact (not a particle and *any* antiparticle, but *its* antiparticle), the net total for any of their conserved quantities (like charge) is zero. That means the major legal interaction is that the two particles are destroy and produce photons. While photons are particles, we tend to think of them as just energy, so the particle-antiparticle interaction is an "annihilation": two particles go in, energy and zero particles come out.
The "its antiparticle" bit is important. You don't see a lot of antielectrons because a free antielectron would easily encounter an electron and annihilate. But there are plenty of antineutrinos because they interact weakly with the rest of the world. An antineutrino interacting with, say, a proton does not cause annihilation. Even an antielectron interacting with, say, a proton doesn't do anything special.
Oh, also, it turns out that, at least for the "normal matter" particles like electrons and protons, the universe seems to contain pretty much only the normal-matter particles and (relatively) no antiparticles. There doesn't seem to be any reason, in physics, for one to be preferred over the other. (It's just that in one region of space, you couldn't have a mixture and also have stable matter.) So that's weird.
This is all a long-winded way of getting to the answer that particles that are their own antiparticles aren't particularly exciting. They all have the property that conserved quantities (at least, those that are negated in antiparticles) are zero. So they all naturally have annihilation interactions: when two collide, they can annihilate and form protons. But the annihilation interaction isn't particularly dramatic or weird, it just sounds interesting. The particles all probably also have interactions with all sorts of other types of particles, too, and it really comes down to what particle it happens to collide with first. Maybe a photon and an antineutrino interact with a proton and form a neutron.
Most of the particles that are their own antiparticles are relatively neutral to normal matter (and consequently, also to normal antimatter). But they're all a very different kind of particle from normal matter. They're things like force-carriers (photons) and muons, and they interact with electrons and protons differently from how electrons and protons interact with each other.
For some real fun, look up Feynman diagrams, a neat way of writing down different legal particle interactions. One axis is space (in one dimension) and one axis is time. Now, any 90-degree rotation of a legal interaction is still a legal interaction.