When a person pulls or pushes a cart, why is it advantageous for their body be tilted forward? This is not a homework question. I attempted to draw a free body diagram for a person pulling or pushing a cart.
Based on Newton's third law, the following forces act on the body of the person:

*

*forward reaction force done by the ground because of friction between the person and the ground.

*downward force (the person's weight) done by the earth.

*backward reaction force done by the cart.

I am wondering why the body of the person must be tilted forward. I have not seen any relationship between this posture with the magnitude of the forces acting on the person.
Could you tell me why the person's body must be tilted forward? How does this posture provide mechanical advantages?

 A: It's similar to why when you pull something with a rope along the ground it can never be vertical. It needs to have a horizontal component to apply a horizontal force.
This is not quite as true with rigid structures like legs, but it similar in that it reduces bending stresses on the leg bones and torques on you knees and hips. It effectively reduces the lever arm formed by our legs between our hips and feet which increases the force that can be applied. A  wheel (a round lever) is an example of where this does snot happen: a straight lever that pushes parallel  to the ground while being perpendicular to it.
If you calculate the internal forces on the leg bones and the force/torque at the joints, you will find the differences. What you're doing right now is similar to calculating the force at the edge of a large and small wheel and being unable to find the difference. The difference is the torque at the shaft.
A: It adds a horizontal component to the vertical force exerted by your legs.
When you're standing straight up, the forces exerted by your legs are straight up and down. Your legs are designed to exert a force to counter your bodyweight and allow you to stand like this, so they're quite strong at countering this force.
When you're leaning over like this, each step you take restores you a leaning version of standing straight position, and it allows you to exert this standing-up force your legs exert in a horizontal direction - in each of these pictures, you can see the person with one leg straight back, and the other bent in front of it; at the completion of each step, you could consider them to be standing with both legs straight back for a brief moment before they move the next leg.
A: Probably the easiest way to analyze it is in terms of torque balance.
So, you already know that a rigid body which happens to have a constant momentum, this constancy requires all of the external forces to sum to zero: this is force balance. (It is sometimes confused with Newton's third law; Newton's third law just says here that “You don't need to consider all the forces, only the external ones. The internal forces cancel each other out.”)
Well, the same thing can be said of any conserved quantity, it doesn't just have to be momentum. If my sink is a bit clogged and there is a standing water level in my sink, conservation of mass of the water is going to guarantee that if the water level is not changing, then water coming in from the faucet is equally balanced by water leaving, either by evaporation or by slipping around the clog. There is a state of water flow balance. All of these balances are called “dynamic equilibrium” conditions.
The conserved quantity that matters in the torque case is called angular momentum, torque is a property of force that can transfer angular momentum, and if we notice that a human is staying constant orientation, then we can conclude that they are in a state of torque balance.
Once you know to look here the rest of the analysis is very straightforward. The forces are roughly comparable: the horizontal component to force on the feet always points forwards when walking forwards and has to provide also the horizontal component to force on the cart. The cart’s reaction force on the person is  thus backwards, the force on the feet is forwards, and they are about the same magnitude. Torque says we should multiply by the lever arm, which is to say the distance from about this person's belly button where the line of the force intersects them. The problem is that the force on the feet is about as far as it can be, whereas if I'm grasping something with my arms at about waist level, that's about as close to my center of mass as it can be. So the torque from my legs might easily be 10 times the torque of the weight that I'm pulling.
The leaning forward allows the person to use gravity to counter the torque.
A: It's about acceleration.

You don't need to pull a load behind you. You see this when you wish to accelerate significantly1 and not fall down.

If you aren't accelerating you shouldn't be leaning (or you'll fall over). At the end of a long enough race you'll be going as fast as you can. So no more accelerating. No more leaning. 2

As you move through a curve you accelerate towards the center of the curve. To avoid falling over you lean into the curve.

If you aren't accelerating you don't need to lean. You can be moving as fast as you like. So long as your speed isn't changing there is no acceleration and so no leaning.

You can prove this to yourself by balancing a stick on your hand. Let it lean in any direction and to return it to balance it you'll have to accelerate it in the direction of the lean.
Do that on a moving train or a flying plane and you'll find the stick behaves the same way. Unless you're experiencing turbulence (which is also acceleration) the stick will behave the same as it did when you were standing still.

But do this in space and the leaning is no longer a thing. That's because you need gravity for leaning to mean anything. It's the pull of gravity that gives you potential energy that you can turn into horizontal kinetic energy as you fall over. When you lean against acceleration you're using that to keep yourself balanced.

It should be understood that leaning doesn't add a continuous source of energy. Rather it lets your legs work both vertically and horizontally. Vertically you lose and recover potential energy as your legs work. The leaning turns that energy into forward acceleration. Horizontally, your legs simply push you forward. Something they can't do if you stand upright. At least, not without walking out from under you.

You may feel the leaning seems more pronounced with the heavy cart. There is a good reason for that. A cart adds mass you can't lean. So you have to lean more for longer to get to the same speed. That's because the added mass adds inertia. That's why leaning while accelerating a heavy cart looks so much more dramatic. Even when the ground is level. With the added mass that you can lean further and have more time before the acceleration ends.
A: You do not want your body to rotate.
The forward friction force wants to rotate your body backwards.
You tilt your center of mass forward so that the normal force supplies a counter-balancing torque that wants to rotate your body forwards. The two torques cancel and your body is in rotational equilibrium.
A: It could not be easier to explain this.
Using your diagrams:
You're trying to move the cart left or right - not up or down.
So ............. you have to push left or right.
It's that simple.
