The energy comes from the car
First of all, the tire is rolling forward, in the direction of traffic. It looks like it's coming back toward the car, because it quickly decelerates, and is not moving as fast as the car in the accident or the vehicle with the camera. But the rear edge of the tire (which the car hits) is moving upward, and then forward.
The air in the tire makes the tire compressible and elastic. Essentially, it's a spring.
When the car hits the tire, the forward roll of the tire and the high point of impact of the car means the tire pulls the car over itself, and the car compresses the tire against the ground. This stores energy in the deformation of the tire, which comes from the car's forward velocity. (The car slows down slightly at this point of the accident.)
As with any object striking a spring, this compression continues until the tire can push back and accelerate the car away from it. At this point, the car is nearly on top of the tire, so the tire is pushing upward. As it expands, it flings the car into the air.
So how much of the car's energy was turned into upward motion? We'd need to know the exact velocity and height of the car to calculate that.
But it looks like the center of the car rose roughly 3 meters, and they're traveling freeway speeds of roughly 100km/h.
Distance = 1/2 times acceleration times the square of time, so the car had enough velocity to rise for 0.6 seconds, which agrees with the video. This means it had an upward velocity of 6m/s (0.6s*9.8m/s/s), or 22km/h.
Energy varies by the square of velocity. This means we can find out how much of the car's forward velocity was turned into upward velocity by
- Squaring the car's forward velocity.
- Subtracting the square of the upward velocity (to represent the redirected energy)
- Taking the square root of the result (to get back to velocity.)
This means that a car traveling 100kmph forward initially would have a 97.5kmph forward velocity after launching off the tire. This took 5% of the car's energy, representing 2.5% of its velocity, to achieve.