I know the engines drive the ship through space, but how does the crew steer it? What is it that Sulu, Chekov, and the other navigators are supposed to be controlling when they maneuver the vessel?
"All right... Everybody lean..."
In-universe answer: At impulse and slower speeds they would use some kind of reaction control system (basically, thrusters) or a small-scale version of the impulse engines. At warp, they would have to alter the shape of the warp field, or drop out of warp, turn as described above, then enter warp again.
Out-of-universe answer: We're used to how airplanes and submarines move, so the way the Enterprise banks and turns looks "natural" to us, when in fact a real spacecraft wouldn't move like that at all. Remember that a lot of the effects shots are based on what looks good, not necessarily what's scientifically accurate.