A plasma with a magnetic field in it will carry that magnetic field around with it as it moves. (The magnetic field acts on the plasma and the plasma acts on the magnetic field--plasma physics is hard.) So far so good--the plasma erupts from the Sun, and carries some of the Sun's magnetic field with it. Think of it stretching those field lines further out into space.
But it turns out there's a lot of turbulence there too, so the plasma sometimes gets whirled around on itself. What happens when the magnetic field lines cross?
You get what you might think of as a magnetic short circuit. The magnetic field "lines" join and shift--quite rapidly. And this happens over a fairly sizeable chunk of space.
A rapidly changing magnetic field produces an electric field (and vice versa, of course). So out in the middle of Nowhereville-By-The-Sun, a bit of the flying plasma experiences a strong electric field. The protons move one way, the electrons another. Most of them bounce off other things and slow down and go back to balancing each other's charge eventually, but some luck out and keep getting accelerated (remember that the reconnection region can be large). Acceleration over a long distance builds up speed.
Some of those protons (electrons tend to get scattered away more easily) head out in our direction, and eventually may hit the Earth's magnetic field and spiral in to be part of an aurora.
The protons of interest here head back to the Sun, where some of them smash into another nucleus. Those kinds of smashes usually produce pions, and the charged pions eventually decay into a muon and a few neutrinos. And we can maybe see some of those neutrinos. They aren't terribly high energy, but they're higher energy than those usually produced in the Sun--maybe 10 times higher.
It's ironic--neutrinos produced in the center of the Sun are ordinary low-ish energy, but those produced on the outside during flares can be quite a bit peppier. I can't think of any everyday examples, but I notice that boiling water in a pot only really splashes you when bubbles reach the surface and pop. The pop can really fling hot water around. Remember boiling something dry too fast back in chemistry class, and getting "bumping?"
We expect similar sorts of things to happen, but on even larger scales, when the blast wave from a supernova crashes into interstellar gas. Some of the neutrinos IceCube detected were over 10,000,000 times more energetic than ordinary solar neutrinos.