Basically the idea is that all of space is filled with waves of various sorts. (This is a quantum-mechanical thing) Normally, in empty space, there are waves of all different wavelengths, but when you have a small space between two surfaces, only the waves whose wavelengths are factors of the size of the space will exist in that space. Any other waves will be reflected back and forth and cancel each other out. (Example: in a 1cm cavity, you could have waves of 1cm, .5cm, .25cm, etc. but not, say, .7cm)
Now, if you have two flat plates sitting a short distance apart, the waves between them are limited to certain wavelengths. But the waves outside the plates can have any wavelength. So you have a higher energy density on one side of each plate than on the other, and whenever there is a difference of energy densities, you get a force that tries to equalize them. That force is responsible for the Casimir effect.