In a crash everything becomes much heavier - weighing its usual weight times however many G's (G = force of gravity) are in the crash. A 30mph crash will have about 20-25G's which means everything will weigh about 20-25 times its usual weight. A 30 pound dog will weigh more than 600 pounds in a 30mph crash and if not properly restrained will fly around the vehicle. Neither you, nor the dog, will do well in the crash if the dog lands on top of you. Imagine what the dog could do to a child if the dog lands on the child in a crash.
Studies with people show that when one person doesn't buckle up in the back seat the people in the car who did buckle up are up to 3 times more likely to die in that same crash because the unbuckled person becomes a human missile flying through the car. So make sure every human in the car is buckled up - and every furry critter is too!
Your dog or cat is more likely to survive a crash when properly restrained. The first job of a restraint is to keep you in the car - as studies show if you are ejected from the car you are 4 times more likely to die in that crash. The next job of a restraint is to try and distribute the crash forces over the strongest areas of the body so that you can withstand the impact with as few injuries as possible.
Sadly, many dogs run off after a crash - usually because they are scared or injured. Crashes are scary enough, but losing your animal - or watching the animal get hit by a car as they run off into the street - makes things so much worse. An animal who is properly restrained will not be able to run away and endanger themselves.
An unrestrained animal can interfere with first responders quickly reaching and tending to injured humans.
If injured, your animal is most likely to get the medical attention they need as quickly as possible if they are easy to locate AND easy to handle. Injured/scared cats or dogs can act in ways that put the adult approaching them at risk - which can lead to the animal not getting the help they need if the adult is afraid of getting bitten or scratched. An animal who is restrained is easy to approach and handle in a way that can keep the person safe from being hurt by the animal.
Unrestrained animals can easily distract the driver - putting everyone in the car at risk. A survey of dog owners found that while driving nearly one-quarter (23%) have used their hands or arms to hold their dog in place while applying brakes, and 19% have used their hands or arms to keep their dog from climbing into the front seat.
An unrestrained animal is also distracting as the driver may be checking to make sure the animal isn't hanging too far out the window, jumping on the baby, trying to jump on the driver's lap, etc.