For ideal rigid surfaces as here, the normal force is an ideal constraint force whose value can only be determined from its effect on the system and NOT from its intrinsic nature.
Now the bead acted on by gravity has two tendencies: (1) to slide a place where the wire is horizontal where gravity will NOT try to make it slide, (2) to torque the helix so that it, the bead, goes lower.
The two tendencies are mutually satisfied by the bead sliding down the helix cranking it in the opposite direction to the animation.
However, the attribution is certainly wrong in an absolute sense since Archimedes' screw was probably known centuries earlier than Archimedes in Babylonia (see Wikipedia: Archimedes' screw: History).
Probably, the ancient Greeks just attributed Archimedes' screw to Archimedes because they did NOT know its real origin, and so attached its invention to a famous celebrity who may have been alive at the time they first learnt of it (see Wikipedia: Archimedes' screw).
On the other hand, it is always possible that Archimedes independently invented Archimedes' screw or at least introduced it himself to the Greco-Roman world.
But Archimedes' screw was certainly invented empirically by people just playing around with screws and pipes perhaps after an observation of an accidental Archimedes' screw.