Explain why your body feels like it is being pushed back when the car starts back up again

Due to inertia.

Explanation:

As per law of inertia, a body will remain in its state of motion until an external force, unbalanced in nature, is applied on it. When the car is at rest, the person sitting in it will also be at rest. When the car suddenly moves, the body resist the motion due to inertia and it stays in initial position while the car moves. Due to this the body will be pushed back.

This push is felt due to inertia

Explanation:

Inertia is the resistance, of any physical object, to any change in its velocity. This property is also called the first law of motion which says a body in rest will remain at rest and a body in motion will stay in motion until an external force acts upon them. The same thing happens when our body is at rest and the car suddenly starts backing up again. Because the rest state of our body experiences a sudden push and our body resist it due to inertia.