Yeah, neither Star Wars nor Star Trek really fit in the realm of "hard" science fiction. Hyperdrive and warp speed are both storytelling devices that allow the characters to travel impossible distances over a short period of time.

But to get "technical" about it, I've always understood hyperdrive to be a way to punch through our universe into another one where the laws of physics as we know them don't apply ("hyperspace"), allowing them to travel much faster than the speed of light would be in our universe with none of the negative effects.