Focus your eyes as if you're looking at a more distant object. You know how if you look at something a little further away and hold your finger in front of your eye it doubles? With a magic eye, the image is repetitive, and you're trying to get the doubles to overlap and "lock" together, tricking your eyes into thinking they've focused correctly. The repetition is varied in such away that it makes the illusion of a 3d object.

The kind of weird thing is that you can achieve the same effect with any repeated pattern - floor tiles, a fence post, etc. But there's no underlying illusiory image, so you just see basically the same thing but everything else is focused wrong.