Can the annihilation of matter and antimatter be explained by the electro-weak interaction?
Can pair-production be explained in the same way?
No, particle creation and annihilation is not an electroweak phenomenon.
It happens because in quantum field theory particles are not fundamental objects. Instead the fundamental objects are quantum fields, and particles are just excitations in these fields. So when you add more energy to a field you get more particles. The number of particles is not a conserved quantity.
This probably sounds rather vague, but I'm not sure how you can make it clearer without getting more technical, and I guessing you want a non-technical answer.
The wiki article is fairly descriptive of particle antiparticle annihilation.
There you will see about proton antiproton annihilation which happens not because of the electroweak force but because of the strong force.
A basic tenet to call an interaction an "annihilation" is that quantum numbers are zeroed. Electron number is zeroed in electron positron annihilation.
Protons have three valence quarks and antiprotons three valence antiquarks and when quark meets antiquark quark number becomes 0, gluons appear and there are hadrons created picking up the slack in energy and balancing quantum numbers in general for the whole interaction. This means mesons appear, which are bound pairs of quark-antiquark. etc.
The Feynman diagrams of quantum electrodynamics and chromodynamics allow one to keep count of what is happening in the interactions and with what probabilities, @Nick Kidman's answer is demonstrating for you.
electron-positron annihilation $e^−+e^+→γ+γ$
and pair production $γ→e^−+e^+$
are represented in Quantum electrodynamics. That means they contain similar computation preseciptions for (propability) amplitudes for transitions like $\langle γ+γ| e^−,e^+\rangle$, $\langle e^−,e^+| γ\rangle$ or more complicated ones. (Shown in the graphs are the lowest order contributions.)
The electroweak theory is concerned with electrons, photons and/or other particles. You'll have to be more specific what matter means you you here. Antiparticles are a general feature of relativistic quantum field theory (of charged particles) and there their occurrence is necessarity.
Here an informal tool to help you figure out what processes your theory at hand will describle:
Whenever you talk quantum electrodynamics, then you're concerned with an electron-photon interaction of the form
$$-e\ \bar \psi (\gamma A) \psi.$$
In this expression $e$ is just the charge and $\gamma$ is some matrix. What is important is to observe that two fermion particles $\psi$ (e.g. electrons $e^-$) and one vector bosons $A$ (the photon $\gamma$) come together. The interactionvertex of quantum electrodynamics is
You see that this is the fundamental ingedient to construct the pictures above.
By this reasoning you get an idea, which processes can be described by a theory. Here is a more wild one:
Now you also know that the Feynman diagram you see on each QFT page on wikipedia in the side bar template is not a diagram of pure quantum electrodynamics (but you see that the vertex for the green curly gluon line is somewhat similar to that of the photon). Here another picture, just because it's fun to post exmaples
You usually want improve the computations by considering more graphs with the same in and outgoing particles. Theoreticans are anxious about graphs with loops like
In any case, you also have to know about phenomena, which are forbidden because of e.g. charge conservation $-$ e.g. $e^-+e^-$ have negative charge and therefore can't annhihilate into photons.
Now to get an impression on what the electro-weak theory can do, you "just" need to take a look the Lagrangian of the theory, find the interaction term, and start drawing pictures. Equivalently, you have to find out what the smallest interaction vertices of the theroy are. And it's also good to know the names of the particles the theory contains.