If you were a rigid metal pole welded to the floor of the bus, there would be no bending / lag of the upper section of the pole moving forward as the bus moves forward; it would all accelerate together with the bus.
So this clearly has something to do with the "floppiness" of people. Therefore let's consider an extreme case.
Imagine an ice block on the floor of the bus. As the bus moves off, the ice block would not move with the bus, but it would instead, to an observer on the bus, appear to slide to the rear.
If the bus had no windows, someone on the bus could make the honest observation of some kind of "backwards force", as you put it, that accelerates the block to the back of the bus.
As a matter of fact, it is clear from an observer on the pavement that the block is in fact remaining completely stationary relative to the road. You can almost imagine it floating in mid air and the bus just moving away from underneath it.
So in this case, due to the lack of friction between the floor of the bus and the ice block, the block does not even notice the bus moving off.
The same thing happens to the upper half of a person's body. Their feat can be considered to be like a rigid metal pole welded to the floor - firmly held in place by a strong friction force. But as you move up the body, this frictional force is transmitted less and less between vertebrae and other bones. By the time we consider our head, it is very weakly supported - just the muscles in our neck.
So it is clear that the further up our body we go, the more our body behaves more like the ice cube. That is to say that it is not being "pushed backwards" by the bus, but merely remaining more and more fixed in place relative to the road.
Of course, we are not just a pile of bones - our muscles are trying their hardest to keep our head and arms above our feat and not lag behind, so to speak. It is now clear why we subconsciously step back, and why people get whiplash. We step back to try and help our muscles - give them more of a structure to push off and keep our hips moving forward with the bus, and people get whiplash when the muscles in the neck aren't strong enough to keep our wobbly head moving along with the rest of our body - the head stays still and our body is jerked forwards/backwards. Note that the head is not experiencing a force when someone gets whiplash, it is experiencing an absence of a force as the person's body gets pulled along by the bus / car in a harsh period of acceleration.