the reality is that even at light speed you wouldn't see any stars move
the visual effects department shows the streaks to communicate that the ship is moving
I think the tongue-in-cheek explanation is that it's not stars but excited dust particles reacting with the navigational deflectors . . .
the view from a ship traveling at FTL speeds would look like a black tunnel with everything behind red shifted, and everything ahead blue shifted
so the most you'd see ahead would be any infrared shifted to visual range, and behind you'd see things like xray and gamma rays shifted down to visual range
you wouldn't see any stars unless you were close enough and only if they were directly ahead or behind . . . none whoooshing past the ship