Why cars rotate during small overlap crash test?

As the car hits the obstacle, the latter exerts a force $F$ on the former. This force is eccentric with respect to the CoG (the small cross in the center of the car schematic) and this causes a torque (moment) $tau$ about the CoG:

$$tau=Fd$$

With Newton's Second Law, the torque causes angular acceleration $alpha$:

$$tau=Ialpha$$

where $I$ is the inertial moment of the car about the CoG.

In other words, the car starts rotating about the CoG.