The force from moving forward is applied at the tires right at the road surface. The car body can move independently of the axles a little because of the suspension. Force applied below, body of car with the majority of the weight above. Obviously the car body will move backwards when the tires move forward.
If it’s not obvious, consider a more exaggerated case where the contact with the ground where force is coming from and centre of mass are even further apart. If you were standing there on a rug and I pulled it forwards, your feet will come forwards with the rug. However, your heads going to go flying backwards. The car does the same thing, it just thankfully has a really low centre of gravity and a lot of weight so it doesn’t tip over. Motorcycles can exceed this though and flip if they accelerate enough.
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