1. Consider a simple idealized case of a vanishingly-small frictionless bead strung on infinitely-thin rigid helical wire.

2. Because the bead is frictionless, the bead can only exert a normal force on the wire and vice versa.
   A normal force is perpendicular to the contact surfaces.

   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.

3. Say also the wire is free to rotate on its axis.

4. First, say we let the bead slide freely without cranking the helix

   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.

5. Second, if the helix is cranked in the direction of the animation, then the bead would rise like the ball in the animation. In this case, the torque of the cranking converts mechanical energy into gravitational potential energy of the bead.

6. The analysis is rather complex even for the idealized case of the bead on the helix mainly because its inherently 3-dimensional.

7. The realistic case with fluids is harder to analyze even just qualitatively, but the qualitative behavior is just as for the case of the bead on the helix.

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.