It looks like the band should work its way off this belt-drive system, but a crowned pulley works opposite of what appears intuitive.
To better demonstrate the principle, I built a jig from lego, using a rubber band and an exaggerated crowned wooden pulley.
The higher section of the crowned pulley puts more tension on the rubber band than the narrower edges. As a result, the rubber band flexes into a slight arch towards the middle. As the rubber band winds onto the pulley, this arch causes the band to always wind further up on the conical section than what was previously wound on. The higher point on the pulley always creates more tension in the belt and causes it to arch in that direction.
With this exaggerated crowned pulley, it takes just a few turns for the rubber band to wander from the edge all the way to up the center hump. Once the rubber band is on top, the maximum tension will be in the middle of it, and it no longer has any reason to arch in either direction.
The mechanism is fairly subtle in most flat belt pulley transmissions. The crowning on a crowned pulley is typically barely noticeable when looking at it. My demonstration uses an exaggerated crown to make it easier to see what happens.
With the much more subtle crowning on a typical pulley, the self-centering of the belt happens more slowly. If the pulleys are misaligned, it may never center itself. Flat belt transmissions require much more precise alignment than V-belts.