I don't understand how an Alcubiere space warp can actually let you cross space in a short time?

I don't either, and I do not really have the time or inclination to find out given all of the other more interesting things I have to do these days... like answer a gazillion questions every day about things I think I can speak meaningfully about!

This method of surfing across space has been looked at very carefully by folks that know general relativity, and the bottom line is that to make it work you would have to do something rash like incinerate all the stars in the cosmos to generate enough energy to create the warp effect. Mother Nature again gives with one hand and takes with the other when it comes to interstellar travel.