If two plates are put partly in waving water parallel to each other, there will be less wavelengths between the plates than elsewhere and this will cause the plates to move towards each other. Why do waves push the plates? Water waves are transverse waves meaning that water molecules move only up and down to a direction which is parallel to the surface of the plates. So why would these water molecules push the plates even though they don't move towards the plates? For longitudinal waves it would be easy to understand, since the molecules would actually push the plates.
There is hydrostatic pressure in the water, so when the slope of water at one side of the plate is different from the other side, the pressure difference on both sides will produce net force.
"Water waves are transverse waves meaning that water molecules move only up and down"
If all the water molecules would move only up and down, the water density would be either changing, or there would be some empty space created somewhere inside the water slope. None of it is happening, so the water molecules need to move sideways also.
I think this is an interesting question. The first question is obviously whether the "Casimir" effect in ordinary fluids is real. There is a nature news column that claims that the whole thing is an old wives tale, but there are also a couple of you tube videos, and a fairly detailed article in the American Journal of Physics that seem to confirm it.
Obviously, nobody claims that this is a quantum effect. It is not due to quantum fluctuations, and not proportional to $\hbar$. If the effect is real, it would be due to an externally driven spectrum of water waves (generated by wind action, thermal noise, or, in an experiment, by a vibrating surface).
Could the effect be real, given what we know about water waves? I think so. Two parallel plates will affect the spectrum of waves in between the plates in much the same way that a conducting surface does in the QED case. Also, water waves carry energy and momentum, and the article cited above attempts to give a theory of the "Casimir" effects based on these facts.
Note that we are considering surface waves in an approximately incompressible fluid. These are distinct from longitudinal sound modes, but I think it is not correct to call them transverse. Purely transverse waves in fluids are diffusive (non-propagating). In surface waves the motion of individual water molecules is approximately a circle, where the radius of the circle decreases with depth. Water waves clearly have a non-zero energy density, but whether they carry linear momentum is more subtle. In the linear theory the motion is exactly sinusoidal, and the momentum density averages to zero. However, this is not the case beyond linear order, and you can find long discussions in the literature on how to compute the linear momentum carried by water waves.