Basic explanation of baryon acoustic oscillations; what if anything is actually oscillating?
tl;dr nothing is oscillating now, but it was in the early universe. Now we just see the frozen relics
This is just an attempt at a simplified explanation of the Wikipedia article linked in the question.
There were pressure waves in the very early universe. They came from a balance of gravity and radiation pressure (the matter at the time was hot enough to be opaque). Whereever there happened to be a locally higher density (of matter, radiation and dark matter) the excess pressure would cause an expansion and then gravity would cause a contraction sending waves out at (according to the article) half the speed of light.
As the universe expanded and cooled, matter eventually became transparent (it was hydrogen and helium gas) letting the radiation pressure escape and leaving higher densities both in the location of the original concentration and at the crest of the wave a certain distance away (but the same certain distance for the first waves from each concentration). Those high densities are where the galaxies formed, so what we look for today is an excess of galaxies at a specific separation and that's what we see.