Imagine the wind as a stream of tennis balls, all moving at close to the same speed. (This is in some ways a poor model for actual wind, where the velocity distribution of the component particles is much wider, and only net pressure matters on the macro scale. But it does simplify things. I'll point out, later, where the simplification might cause some trouble.)
What you're saying is, you want to see, from the ground reference frame, forward-moving tennis balls (wind) catching up to and hitting some part of the cart, being slowed down and so providing a push to make it go faster or at least maintain speed.
Except for the part about "catching up" which will need to be refined a bit, this can be done.
So, we start by building a cart with a big tennis racket mounted in the back, face-on to the wind. Put this cart into the tennis-ball current, and it will start moving, as tennis balls hit the racket.
But, when it gets close to wind speed, fewer tennis balls per unit time catch up to the racket and hit it, and when they do the relative speed is very small so they no longer give it much of a push. When the total net force is no longer greater than frictional losses, the cart won't go any faster. If you were to push the cart to wind speed, no tennis balls would hit the racket at all, and if you pushed it faster, tennis balls would start hitting the front of the racket and slowing the cart down.
The way to get around this involves two factors: one, the face of the racket has to be placed at an angle; and two, the racket has to be moving laterally relative to the cart; that is, in a plane perpendicular to the cart's trajectory.
We could rig some kind of mechanical forehand-arm to sweep the tennis racket side to side across the back of the cart while holding it at one angle when going left to right and a symmetrical angle when going right to left. But it's easier to just spin the racket around by the handle (on an axis perpendicular to the handle). If that sounds a bit unbalanced, we can start with a two-headed racket, with a head at each end of the handle, and spin it by the middle of the handle. The axis is perpendicular to the handle, and oriented parallel to the cart's trajectory.
That takes care of the motion, now we must consider the angle. One possible angle (call it zero degrees) is for the racket heads to be flat in the plane that they're spinning in. But that gives us no advantage (or any real difference at all) from the single stationary racket.
Another possibility is for the racket heads to be turned 90 degrees from the previous case. That turns the device into essentially a sideways paddle wheel. That would also be ineffective: it would bat any tennis balls that caught up with the cart (or that the cart caught up with) off in all different directions but it would only ever accelerate them in the plane perpendicular to the cart's trajectory, giving no benefit regardless of the apparent wind speed.
So instead, we twist the two heads of the racket so that each is at an angle in between those two cases, say 30 degrees from the first (flat) case. And we set the direction of spin so that the leading edge is toward the front of the cart.
The spinning racket now resembles a propeller; this is not, obviously, a coincidence.
We drive the propeller from the wheels as in the DWFTTW carts, so that the propeller speed is proportional to the cart's ground speed. The gear ratio has to be "correct" relative to the wheel size, the length of the "handles" of the tennis-racket propeller, and the angle of the racket heads. I won't go into details about what makes the ratio correct -- instead, I'll just say that IF the ratio is in fact correct, the following description of what happens will apply.
Starting out at low speed, we have, as before, tennis balls in our tennis ball current ("wind") hitting the back of the rackets. There are a few differences from before: the rackets are at an angle instead of flat on, and are moving laterally. As a result, when a tennis ball hits a racket, it's going to bounce off at an angle instead of straight back. This will give the cart a little less push than before with the flat racket. But, as long as the ball bounces off the racket with a slower forward component to its velocity, some forward momentum will still be transferred to the cart. So the cart can still be pushed to a faster speed by the tennis ball current.
When we start getting near wind speed, things start to look a lot different than before. When the cart is moving near wind speed, a tennis ball will take a lot of time to catch up to the cart. But when it does, it encounters a racket that is moving fairly quickly laterally, at an angle. As a result, instead of just weakly bouncing off as it would off a flat racket fixed to the cart, the ball will still be deflected energetically. As before it won't be driven straight back but as long as the forward component of its motion relative to the ground is reduced after the "shot", some forward momentum has been transferred to the cart by the collision. In the cart's reference frame, if we were riding the cart, we see the ball slowly approaching the racket, then being driven mostly backward (but off at an angle) at a much faster speed. From the ground reference frame, we would see the same collision as the ball zipping along at "wind" speed, hitting the racket, and being slowed way down while being deflected off to one side or another. That's the "wind" pushing on the cart, even though the cart is already close to wind speed.
But, can this still work at faster than wind speed? At first glance, you might think no, because the tennis balls in the current can no longer "catch up" to the tennis rackets. Instead, the cart and the tennis rackets are catching up to and overtaking tennis balls.
But that's actually not a problem. "Catching up" was never really an issue in the first place, it just seemed that way. What matters is the current, not what any given individual tennis ball in it does. Sure, once the cart is past wind speed, a ball behind the cart can never catch up to it and so will have no effect on it. But the tennis balls the cart catches up to still matter. Because they're still moving forward (relative to the ground), they can still bounce off the back sides of the spinning rackets and propel the cart forward.
Imagine you're riding one of those tennis balls. The cart is behind you moving faster than the "wind" speed (faster than you), then the cart starts passing you. If the rackets were flat or not moving, you could only get hit by them from behind, slowing the cart down. But they're at an angle and moving laterally. So what happens, if you're (un?)lucky, is that the leading edge of one of the rackets cuts in front of you while its trailing edge is still behind you. It then sweeps laterally across your path fast enough that, even though you're going slower than the cart, you collide with the racket before you have a chance to fall behind the trailing edge. As before in the just below wind speed case, your tennis ball is accelerated backward, transferring forward momentum to the cart. (From the ground reference frame, your tennis ball is slowed down in the forward direction, and deflected at some angle.)
This only happens, as I said, with the right combinations of racket angle, racket lateral speed, and relative cart speed. If the racket is too "flat" or not moving laterally (spinning) fast enough, then when the leading edge cuts in front of you you might fall behind the trailing edge before the racket can hit you. If the angle is too great, the ricochet when you do hit might not slow your tennis ball's forward velocity.
Does this help explain it? It amounts to the same thing as the vector diagrams but some people have an easier time visualizing discrete interactive bodies than vectored currents and forces.
One area where the simplified model gets in the way is that if you imagine the tennis-ball cart going exactly at tennis-ball speed, it seems it would "use up" the tennis balls that happened to be traveling with it, and be left in a vacuum! This doesn't happen with actual air, because real wind is a net current in which the individual particles are always moving at all different speeds and directions, so the cart at wind speed cannot run out of air molecules.
Respectfully,
Myriad
(apologies to Dr. K 
)
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