Also, don't get hung up about how a propeller works.
The details of how a propeller works, also, are not important to understanding how the cart works.
You already said that you understand that a propeller can accelerate a vehicle whether the air is still or moving. That's what the propeller is doing for the cart.
Think of the cart having an engine onboard to power the propeller. You can see how the propeller could make the cart move. As long as the wind isn't so strong that it knocks the cart over or blows it backwards or whatever, the propeller will be able to drive the cart forward.
For the DDWFTTW cart, replace the engine with a connection from the wheels to the propeller so that the turning of the wheels drives the propeller. The wheels cause the propeller to spin. The spinning propeller moves the cart forward.
-- Roger