Star Wars approaches the problem through the slightly silly "accelerate to light speed and then we can punch through to hyperspace where relativity does not apply" route, which is a bit dubious.
Star Trek had a slightly better solution (at least, once Okuda and Sternbach took over techy stuff).
The warp field serves two functions:
(i) Push the ship into subspace, a 'layer' that covers the same volume as normal space but... using less space. Yeah, I know.
(ii) Shield the ship from the effects of relativity.
This is handy because it allows FTL travel time without FTL travel, and allows energy weapons to be fired between ships at warp, etc etc.
Problem is they never explained how it was meant to work, but assuming subspace (or something matching the description) exists, and we can find a way to evade relativistic effects, AND we don't destroy ourselves along the way, someone will figure it out eventually.