For some kind of tech explanation, the deflector reaches out ahead of the ship. It pushes to the side particulate matter, cosmic rays and free hydrogen, so that the ship doesn't hit these things at hundreds of times the speed of light. Basically tunnels a hole that the ship flys down. But the warp drive field extends hundreds of meters, maybe several kilometer to the sides of the ship, all this stuff pushed to the side by the deflector passes through the warp drive field and that's what we see as streaks, the matter interacting with the structure of warp drive field. Outside the field it's warp eight, inside it's slower.
This explanation doesn't work for Where No One Has Gone Before
unless the field is really really big.