What does caught in the headlights eyes mean?

Answer 1

WHen deer are in the dark say by the side of the road and a car's headlights shine on them, they are blinded momentarily and are susceptible to being shot. That why this method of deer hunting is against the law. It is an experession that means that you are too stunned to move. In other words you are terrified. It comes from when you catch a deer in your headlights and they just freeze.

Answer 2

Deer and other animals, if caught while standing on a road by the headlights of a car, will instinctively freeze. They will stand stock still and stare at the headlights as if they are mesmerized by them.

"Caught in the headlight" eyes mean a frozen, stunned, mesmerized look.