There are a few (vague) descriptions of how the drive works. Nothing concrete, unfortunately
It has something to do with space being quantized
André: ... We have a faster-than-light drive system, which we call quantum drive as opposed to warp drive (the idea being that
space, at the deepest level, is quantized, like the energy levels in
atoms, and if someday we understand how this works we might be able to
manipulate the fabric of space to travel faster than light).
Our power source is called Dysonium, a fictional transuranic element we named in honor of the physicist Freeman Dyson.
Aliens and AI: André Bormanis Explores the Science Behind 'The Orville
The shape of the rings is important, they're tori, not just semi-circles
Bormanis has written a whole show bible about the various science
elements within the universe of The Orville. A good amount is
dedicated to the ship. On The Orville you don’t have a warp drive, but
you do have a “quantum drive.” And for this faster than light sci-fi
ship they are using the theories of physicist Miguel Alcubierre who
has postulated a form of drive that works within Einstein’s general
relativity. And this theory’s call for torus rings are what is behind
the rings of the U.S.S. Orville itself. They have even named the
exotic matter that is required for such a drive “Dysonium” after
physicist Freeman Dyson.
6 Takeaways From A Set Visit To ‘The Orville’
Each of the rings represents an engine. Each engine contains something that requires containment. Presumably that something (dysonium?) is traveling around the rings at high speed.
Newton: Sir, we've lost both quantum phase inducers in the upper engine! I need to divert all available power to the containment regulators now!
The Orville #10