Why do wet plates stick together with a relatively high force?

This has turned out to be a very interesting question indeed. Based on all the good comments I will amend my answer here. There are (at least) two parts to this problem. One is why the plates stick together in the first place and the other is the exact mechanism by which the adhesion between them is broken when you pry them apart. My original answer dealt with what makes them stick together (the adhesion mechanism) but there's a lot of interesting physics having to do with what happens when you do pull hard enough to separate the plates. There are a number of simple and fun experiments that could be done to put some light on what's going on here. First, you could repeat the experiment with the plates immersed in water so any effects having to do with wet/dewet dynamics are eliminated. Second, you can do the experiment in air but with detergent-spiked water, to greatly reduce surface tension effects. Third, you could perform the experiment with transparent glass plates or better yet sheets of flat glass and record what happens with a digital camera as they are separated. I now return you to the regularly-scheduled program.

When a set of identical plates is stacked together, their adjacent top and bottom surfaces fit together well with very little gap space between them. If we try to pull them apart, it is easy for air to flow into the gap and hence allow that gap to grow, and for the plates to come apart.

If there is water in that gap, and we try to pull them apart, several things happen: first, we have to retract the thin film of water into a glob in the center of the plate stack, and the viscosity of the water resists that deformation.

Second, the adhesion that the water molecules have for the surfaces of the plates makes the water want to stay in contact with those surfaces and not get sucked back into a glob. It takes work to de-wet the plate surfaces, and so work must be applied in order to pull the plates apart.

In essence, the water acts as a (lousy) glue, but a good enough glue to illustrate several of the properties of a good glue: 1) it has to develop a high viscosity after being put into a gap, and 2) it has to completely wet the surfaces of the gap before its viscosity starts climbing.

The other answers here offer a lot of explanations for general cases in physics, but there is another factor that should be considered in your specific example: suction.

If you wash your dishes with hot water, then you will naturally heat up the plates in the process. After stacking your plates, the air trapped between them will absorb some of that heat as the plates settle, expand as per the ideal gas law, and push some of the air out into the surrounding atmosphere.

After several hours of cooling, the air trapped between the plates will shrink, exerting less pressure on the plates. At that point, the surrounding air will tend to push the plates together, similar to a suction cup.

The 'release' of the two plates is the same effect you see when opening a new jar of jam; the "popping" of the lid is caused by the spring action being released as air is able to re-pressurize the container. The low-pressure state is caused by packing the contents while hot, and allowing the cooling to "suck" the lid down, forming an air-tight seal which keeps the food fresh for years.

Notably, this is an easily testable theory: next time you wash your plates, run them under cold water for a short time before drying. Leave them for the same amount of time, then attempt to separate them. If the plates are easier to separate, then the suction from the shrinking air is a significant factor. If they are not, then the suction is not a significant factor (or the plates weren't cooled enough).

They don't stick together for all liquids

If you were to put a liquid with a large wetting angle (such as Hg which "beads" up), the plates will be pushed apart. For a liquid such as water that tends to spread on a plate, the forces become very large at small separations.

The force depends on the surface tension of the liquid, the wetting angle at the plate, the volume of the liquid, and the separation of the plates.

This can be worked out from capillary theory with the calculus of variations:

Here is a paper that shows the capillary force between two plates as a function of how well the liquid "wets" the plates: https://www.sciencedirect.com/science/article/abs/pii/0001616088903288