Bro, sala liao ley.

In front curtain sync for moving image with own light source(eg car), your image will look odd cos the car will have light trail in front.

In rear curtain sync for the same car, it looks correct cos it will be car followed by light trail.
you're right. However, that's only an understanding of what you get, not how it works.
What happens is if you flash, the camera will take a "sharp" image. (because the light exposes the subject better..)

1) if front curtain sync, the flash will flash first and capture it. Because the rest of the body is dark, and typically require a long shutter speed to expose it while the headlights do not, your sensor will continue to capture the light streaks.
Resultant: Sharp image of the car at the starting position, and the lights streak ahead because there's not enough light to expose the rest of the body.

<<-----car direction
taking -------- to be the light streaks..


2) rear curtain sync, the shutter opens, and the car body cannot be exposed properly. As such, only the headlights are captured. Thus, it'll leave a trail.. When the shutter is about to close, the flash will pop, and expose the car body correctly. As such you will get the car at the end of the motion, and with light streaks behind it..

<<-----car direction

taking -------- to be the light streaks..