In the original Star Trek, that Alcubierre was inspired by, it wasn't explained at all. You just had warp engines and impulse engines. Warp engines made it so the ship could go at warp speed, but go too fast, and they could come off the ship, or the ship would explode.

It was later series that tried to have an explanation for how they worked.

Although I don't think the writers cared particularly much for whether they were theoretically possible or not, anyway. They work through subspace, and that doesn't exist in reality, so a lot of oddities could just be brushed under that.