Part of how your brain does depth perception is by comparing the two different images from your eyes and finding differences in how things overlap. Close things need to shift more to overlap perfectly and far things need to shift less.
Now this all happens automatically and your brain doesn’t actually care what the overlapping things look like. It just tries to find matching sections.
Autostereograms use a bit of trickiness that uses repeating patterns to avoid needing two different images. The brain sees a bit from your left eye that looks similar to a thing from your right eye. But they are different sections of the repeating pattern. But the brain doesn’t know that subconsciously so the depth perception logic just assumes that the two different sections are really the same section cause it’s not paid enough to care.
From there it’s carefully shifting different parts of the repeating pattern left and right to mimick how much shift there would be in a normal image.
Diagram for clarity:
http://www.techmind.org/stereo/geometry.gif
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