Still catching up here. I know this was a while ago, but here's my ( very late )response.
Warp drive "shrinks" space immediately around the ship. Quick example:
You're driving a car down a straight stretch of road that's 50 miles long ( say in Nebraska or Kansas ). The warp drive would be a device that allows you to "compress" the road immediately around you ( but not you or the car )by a factor of, let's say, 10. With it off, you can cross the distance in 50 minutes if you're going at 60 mph. With it on, you cross it in 5 minutes with the odometer staying on 60.
Back to Trek, this is what warp drive does on a much larger scale. The ship, to an outside observer, crosses the distance at faster-than-light speeds while avoiding trouble from the Law of Relativity in its local space ( if you measured the speed of the interstellar gas and dust passing through the warp field for an "airspeed" reading, they'd be passing at far less than light speed ).
Here's the catch, and the potential cause for weathering on ships in Trek. The interstellar material's density is increased proportionate to the compression caused by the field. Even the best deflector device won't be 100% effective, so something will occasionally hit the hull. Minimizing the damage from these occasional hits is the reason for the aerodynamic shape of the ships in Trek, even if they never enter an atmosphere like the Voyager.
By the way, this is based on what Gene Roddenberry originally had in mind, and it was enough to impress Isaac Azimov, who stated that the idea of Warp was the only theory for faster-than-light travel that he'd ever encountered that used known physics to cheat Einstein. Since TNG, the franchise has changed things to make them more fantastic
.
Ye gods, I need a life!