Most of the gibberish is coming from the attempt to explain a magnon and a spinwave to a wide audience in a few words.
Apologies if this is already familiar, but... magnetic materials are magnetic because the atoms in them have unpaired electron spins, such that the atoms can be thought of as (crudely) like little bar magnets. You get a ferromagnetic material when all these atom magnets line up the same way -- that's iron. Now, if you imagine having a row of these magnets standing in a line, like pickets in a fence. If you try to tip one of the pickets over, it'll pull on the surrounding ones and you'll get a region in which the spin direction is no longer vertical, that is, a region where there is a spin deviation. These spin deviations follow a wave form, and that is a spinwave. A magnon is the quantum of excitation in a spinwave. That is, the discrete amount of energy that the spin structure absorbs when the spin wave energy increases. Just like atoms absorb discrete amounts of energy when you have electronic transitions.
For comparison, a photon is a quantum of excitation of an electromagnetic wave.
Spin waves, which are bosons, are very different from electrons, which are fermions. They do not have a net charge, and they can be antiferromagnetic, which makes them on the whole less susceptible to magnetic fields than you might think. Probably quite a few years away from mainstream tech usage (the article reads a bit like justification for some pretty basic research), but so was quantum computing not that long ago.
It's actually not gibberish. It's solid state physics. The same field of science that have you the transistor.