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.