The effect in the original photo is fairly common when panning with fast moving subjects using a slow shutter speeds and wide-angle lenses
. There are no special lenses or tricks required, and it doesn't require a shallow depth of field, but doing it well does require practice.
There are a couple of contributing factors to the effect. I believe the major factor is because in order to fill the frame with the subject using a wide-angle lens, you need to be standing much closer to it
than with a telephoto lens. When it zooms past you, the closer you are standing to it the more quickly you have to spin your body to pan with it effectively. The angle of the subject (which is driving in a straight line) changes relative to the angle of the camera (which is rotating), yet the focal point in the subject (ie drivers helmet) remains stationary relative to the frame. This gives an effect of the subject rotating slightly, which ends up looking like zoom creep motion blur.
Of course, in the photo shown above the car isn't driving in a straight line at all, but is cornering. Nevertheless, the principle is the same. Any changes in distance between any part of the subject and the camera will appear as blur given a slow shutter speed, and the fact that the camera is very close to the subject means that these changes in depth are a lot more obvious (wide-angles tend to exaggerate depth, while telephotos tend to compress it). Dispite the car turning around the corner, it's nose is
moving relative to the camera, as is the rear wing.
There are also other factors involved here (the angle of the camera relative to the horizon, the moment in time the shot was actually taken, etc)... but I'd suggest experimentation is the best way to really understand how it all works. I must admit I haven't tried to delve any more deeply into it.
Below are a few cycling examples of my own that might help. I estimate these cyclists travel at about 60 km/h around the track, which is probably not too different to the speed of that race car at the apex of that tight corner.
1. This was shot at 20mm, f/5.6 with a 1/80th sec shutter speed.
2. Again at 20mm, f/5.6, but this time 1/100th sec. The four cyclists along the bottom of the track are all travelling at the same speed, dispite three of them having motion blur.
3. This is a telephoto shot to provide a contrast to the other 2 shots; showing none of the effect described while still panning with the subject. Shot at 200mm, f/2.8, 1/160th sec.
I should point out that in order to pan with these wide-angle shots, I was spinning around pretty much as quickly as I was physically able to as they rushed past me. The trick is to pick up a point on one of the cyclists in the viewfinder before they get close to you, and then track them as best you can without shooting as they approach. When they get close, just squeeze the shutter (burst mode) try your best to keep locked on to your point. It happens to fast that there's no time to really think or adjust your technique while the shots are being taken, but with practice you can improve dramatically. It is also a LOT of fun being so close to the action like that.