Overview
A Hyperdrive, named after the prefix for shapes extruded into the 4th dimension, is a form of propulsion that utilizes extradimensional forces to accelerate a ship to high velocities. A form of exotic matter exists that has slightly less attraction to membranes, formed in the birth of new membranes and ejected into the Continuum and surrounding membranes. Such matter attracts matter of its own kind but repels matter and membranes slightly which creates a fine gap between its membrane and itself.
Fuel
Exotic matter be harvested from intergalactic gases using large gravitational scoops and processed into pure exotic matter that can be used to fuel Hyperdrives, and once that's done, exotic matter in the fuel tanks will pull more exotic matter from the Continuum, allowing the drive to passively refuel itself. A tank cannot be capped off as that would require four dimensional creation and engineering beyond current technology, and this leads to the tank always pulling in more exotic matter. If it were to pull in too much, it would burst, so hyperdrive tanks have to be vented occasionally to prevent a massive pressure explosion.
Physics
Since an exotic matter field pulls on a ship with equal force, it can accelerates it at a massive speed without the repercussions of typical high-acceleration propulsion. As the exotic matter repels matter, it acts like a buffer between oncoming debris and the ship it carries. This is extremely useful as military grade Hyperdrives are capable of tens of thousands of Gs of acceleration, and such acceleration would make any sort of interplanetary debris deadly.
The current understanding of Hyperdrives is that when an exotic matter field is created, it lifts matter up off the membrane slightly, creating massive force from the attraction of the Membrane that can be converted into acceleration by weakening certain areas of the exotic field, similar to a maglev. Imagine a flat magnetic object above a metal sheet being suspended by a layer of air. The magnet will accelerate very fast to reach a low density pocket of air, and will keep accelerating if that pocket of air is always escaping.