Then what is spinning the wheels?
A quick summary of the basic forces involved are: The wind pushes the propeller, the propeller pushes the wheels, the wheels push the ground, the ground pushes the wheels, the wheels push the propeller, and the propeller pushes the wind. Or perhaps it's better to say that everything is dynamically interacting with everything else.
I think that what this all adds up to is that, by introducing this mechanical linkage, the cart is extracting energy from the difference between the wind speed and the ground speed. This runs counter to our intuition, which assumes that it should be getting its energy from the difference between the wind's speed and its own speed.
That's what allows it to go faster than the wind. If it were based off the difference between the wind speed and the cart speed, then the forward force would go to zero as the cart's speed approaches the wind speed. But the difference between the wind speed and the ground speed is not related to how fast the cart itself is moving. So the forward force on it doesn't disappear as it approaches the speed of the wind, and instead it will continue to accelerate until the forward force balances with drag and rolling friction.
A kite might be a good starting intuition pump here? Kites generally don't do much of anything useful unless they're mechanically tethered to the ground.
From there move on to sailboats. The reason why this cart can go downwind faster than the wind isn't all that far off from the reason a sailboat can sail into the wind (albeit not directly into the wind), but, critically, only if it has a keel or centerboard.
(Edit: Another detail to point out is that the linkage causes the propeller to turn in the opposite direction it would if it were spinning freely. And remember that Newton's third law works in both directions at the same time.)
(Edit again -- Another observation that might help change one's intuition is that, while the cart as a whole may be moving relative to the ground, the part of the cart that's currently touching the ground at any given moment is more-or-less stationary with respect to the ground. And is also mechanically linked to the propeller.)
The propeller arrangement is a glider that is being pushed by the wind. The glider is tethered to a cart whose wheels in-turn drives the propeller to become a "powered" glider. Without a tether, the relative speed of the glider with respect to the wind would be zero. With the tether powering the propeller, the relative speed of the glider is greater than zero, which makes the cart go faster than the wind.
Thinking about the "glider" and the tether as two separate systems makes more intuitive sense, imo.
I think my intuition there came from a boat going the same direction as the wind, say, north at 1 meter/sec. now the boat can't go faster than 1 m/sec, but if you turn 45 degrees to the east you can travel sqrt(2) m/sec, because you're still going north at 1/ms and the boat gets pushed along to the east as well.
the closer you get to perpendicular to wind, the faster you can go.
I still don't quite get sailing into the wind, but I'd imagine this wheel and propeller system works the same way. The vehicle motion restricts moving to just a line, the wind pushes along the x axis, so the vehicle can sort of zoom along in the y direction, much faster than moving along x.
__edit__
oh, I see, it's the difference between ground and wind. a sailboat couldn't do this but something clever with propellers and linkages probably could
The key to understanding this is: sails are devices that work like wings, via lift, not via drag (exception, spinnakers and such). A sailboat has two wings: the sail, and the keel (or centerboard). The boat is extracting energy via how the lift vector of the sail projects onto the vector the keel keeps the boat tracking along. As you sail downwind the apparent direction of the wind begins to rotate forward in response to the boat's forward momentum. With extreme performance boats, this process can continue to the point where the boat is experiencing a forward apparent wind while sailing downwind.
This is all pretty counter intuitive but valid physics. Speaking for myself, the most easy way to get an intuition for it all is to rent a windsurfer for an afternoon. It'll all make sense after that.
People have tried sailboats with propeller rigs like this cart, but they end up being more trouble than they're worth. The current best anyone has figured out is solid wingsails, which have anywhere from 3x to 7x the lift drag ratio of traditional cloth sails. Hence their use in the crazy billionaire bragging contest races.
I blasted through this video pretty quickly, but it appears to explain things well, including how it's possible for a sailboat to exceed wind speed: https://www.youtube.com/watch?v=jJtvGF8vZbE
The cart adds some moving parts, but I am pretty sure that all they're doing is ensuring that the blades' angle of attack is about the same as it would be for a sailboat's sail when it's sailing into the wind, by turning the propeller at an appropriate speed.
The first time comes from the sail, you divide the wind into a component flowing along the sail (doing nothing), and one that is perpendicular to it (pushing it)
Then you take the perpendicular-to-the-sail part and divide that into a side-ways (drift) and a forwards component (propulsive). The side ways one is neutralized by the keel.
Btw I kinda wonder if a rotatable keel tuned just right would allow sailing right into the wind...
After that is where I start to get more fuzzy on the matter. On the one hand it makes sense that by using some of the speed from the wind to do other things, you can extract more total energy from the wind. And by doing propulsion as that "other thing" it makes sense you get up to a higher speed.
But then it seems to me like there would be a problem once you go faster than the wind, because it's no longer pushing you. I'm not sure if some other effect takes over, or if I'm thinking about the wind in an incorrect way, or if my whole line of thought about this is wrong.