These drives are just "things" in spore that are not described at all throughout the game. There is no description about how these drives work and what they run on.
A real way to travel long distances in space is a drive I made up that's kinda like a jet engine, except backwards.
See, there's a long hollow tube. At one end there is a high energy laser pointer inside. the other side is a solar sail like coverage that is pushed by the laser, ultimately tugging the vessel around.