It took me some time to find out how to see these. I'll try to explain how I think it works, in the hope that it might help others.

The first step is staring at a point which would be some distance behind the display. Autostereograms are almost periodic, so you need to trick your brain into adjusting your eyes with an offset of one occurrence of the period. The important thing is that this only works with the stereogram, and that you should not look at the borders lest your find yourself looking at the rest of the page at which point your brain would directly revert to its normal way of seeing things. To get your brain to adjust correctly, you should try, while staring at something behind the screen, to pay attention to what is going on at the stereogram level, and try to look at a point which is closer or further to the screen in order to reach the one period offset in the stereogram.

At this point, you should begin to see something happening at the stereogram level, but it should still be blurry. Resist the urge to look at it, or it will vanish. Instead, keep staring at the point behind the screen, but try to adjust. The problem here is that your brain thinks you are looking at something behind the screen, and makes your lens adjust as if the object you're staring at were behind your screen, which makes the stereogram blurry because it is too close. You can override that, but it isn't easy. The hard part is that you need to keep your eyes staring at the same point while focusing with your lens.

While you're doing that, you may find that you have to start over when you blink. This can be corrected, but it takes some training. Likewise, training will make you able to move your eye around and look at the borders of the image without getting distracted by what is beyond, or move your head around and see how it affects your vision of the hidden image.

Once you become used to autostereograms, you will find that several periodic images can be seen as stereograms. When you look at one of those, there is no hidden image, but you still get a weird effect if you stare at them and move your head around slightly.