Here’s a layperson’s explanation of how to make a beam of neutrinos, roughly speaking (the details depend on the individual experimental facility.)
First, make a beam of protons, just the way you would if you were wanting to load up the Large Hadron Collider. (That’s a story of its own, but I’ll take the proton beam for granted here.)
Next, smash the proton beam into a “target”, just a thin slab of material. The protons will hit atomic nuclei in the material and shatter them, not only breaking them apart into their protons and neutrons but creating many other particles in the process, including pions (examples of hadrons) of both positive and negative electric charge. All of these particles come flying out the back of the target slab, giving us a beam of protons, neutrons, pions, and a few other stray particles.
Now put the beam near a magnet. A magnet will cause the paths of charged particles to bend. The direction of bending depends on the particle’s electric charge; the amount of bending depends on the particle’s energy. So the neutrons go straight on; the negatively charged pions bend one way; and the protons and positively charged pions bend the other way. Let most of these particles just run into the wall; where you leave a doorway, the particles that go through will have roughly similar energies and the same electric charge. In this way, by putting the doorway in the right place, you can get a beam of mostly positively charged pions with similar energies.
The pions will begin to decay, one by one turning into an anti-muon and a neutrino. Before long, your beam has positively charged muons, with a few as-yet-undecayed pions and stray protons left over, and neutrinos.
Now put the beam near another magnet. The neutrinos, being electrically neutral, go straight ahead. The positively charged particles — the muons, and any leftover pions and protons, will bend to one side. Let them run into the wall. What remains? A neutrino beam. Not a very narrow beam, to be sure, but, if you started with a lot of protons, a very powerful one.
By controlling the directions of the initial protons and the intermediate pions, you can get this beam to go in any direction you choose. For instance, you can build the beam at CERN, and point it at Gran Sasso and the OPERA experiment. Not a narrow beam at all — 2 kilometers wide by the time it travels the 730 kilometers to Gran Sasso. But it’s enough to do the job.