How does Propeller windmilling work?
You can think of the propeller as a screw. Take a screw, a screwdriver, and starts screwing. The threads of the screw drive the screw into the block of wood.

The propeller works in a same way: the engine drives the propeller that "screws" through the air.

It also works the other way around. Well kind of, with the screw, because of the high friction btw the block of wood and the threads of the screw, but forget about the friction for a while, if you can. 

If you stop twisting the handle of the screwdriver, and instead push the block of wood against the screw + screwdriver combo, what should happen? The screw should rotate just as if you were screwing it with the screwdriver, as the threads are still sinking deeper into the block of wood.  The screwdriver would of course rotate with it.

Again, it's the same with propellers (and engines). If you have speed, and you shut down the engine, the airflow will push against the propeller just as the (magically frictionless) wood against the screw. The propeller will keep rotating in the same direction, and as it does so, it keeps the engine turning also.

Now, in the case of you no longer twisting the screwdriver handle or the engine driving the propeller, there is no longer a force driving the screw or the propeller through the medium they are engulfed in. Instead, the medium (wood or air) is imposing a force on the screw &amp; the screwdriver, or the propeller &amp; engine to keep them turning.

If you do a google image search with words "propeller tip contrail" you'll find pictures displaying nicely how the propellers are actually screwing their path through the air.