This is an oddly common question, and one I had not really looked into or thought much about until others asked me. The foot first method has a strong symbolism to it, but it also has a very practical reason.
We walk with our feet, as obvious as that might be, so we see the feet as the mode and direction of traveling. Our feet point the way we go and not only take our body but lead it there. The feet first of the coffin or stretcher is designed to mimic this natural movement. The body will have its feet point towards the front of the car, as that is the direction the car is going and the direction the body should be going.
Feet first is also symbolic of life, the transition from "this" life into death or the "next" life depending on beliefs. It is symbolic of the transition from life to death, our passage through life to its end. So feet first holds a lot of symbolism and iconography to it, from simply mimicking the way a person would naturally travel to the passage of life itself.
But it also has a very practical side, and this might have been the origin for creating this practice. Many things within the industry are wrapped in romantic ideals and symbolism, but actually have very practical reasons. This is no different for the feet first procedure.
An important practicality of feet first is identification, especially on a transfer for a number of reasons. The head end of the stretcher is where the zipper starts, and body bags also have the zipper start at the head end. Which means one need only unzip a small portion to see the face, much nicer and easier than unzipping the whole thing. Also when in a fridge or in a car the body will be feet first, making it easy to see the face and reach the wrist tage for identifying the body.
It is also much easier to lift the foot end as the torso is what has most of the weight. Which means the foot end is rather light, making it easier and safer to lift when needed. Once the foot end is lifted into place getting the heavier head end in is much easier. For example, on the transfer the crew can lift the foot end into the car, then push the head end the rest of the way. They can focus on just pushing the head end the rest of the way and not worrying about also lifting it as the foot is resting on the car.
This is the same with the coffin on the funeral, when going up or down stairs it is much easier and safer to go feet first. The ligher feet can be used to essentially pull the heavy head end into place. It is also easier to direct the coffin with the light foot end, to guide it into place and have the head follow.
You can experiment with this yourself, simply go to the shops and place a bag of potatoes in the trolley. Placing it at the front wil make the trolley harder to control and takes more effort to push than placing it at the back. This is why shopping trollies are deliberately designed to have more stuff at the back than the front. The design means weight will naturally move towards the back of the trolley and make it easier to control without realising.
So there you have it, the body moves feet first because of symbolism and out of practicality.